완비 거리 공간 - 위키백과, 우리 모두의 백과사전

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id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <button aria-controls="toc-정의-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>정의 하위섹션 토글하기</span> </button> <ul id="toc-정의-sublist" class="vector-toc-list"> <li id="toc-코시_열" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#코시_열"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>코시 열</span> </div> </a> <ul id="toc-코시_열-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-확대_상수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#확대_상수"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>확대 상수</span> </div> </a> <ul id="toc-확대_상수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-카리스티_사상과_칸난_사상" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#카리스티_사상과_칸난_사상"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>카리스티 사상과 칸난 사상</span> </div> </a> <ul id="toc-카리스티_사상과_칸난_사상-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-완비_거리_공간" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#완비_거리_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>완비 거리 공간</span> </div> </a> <ul id="toc-완비_거리_공간-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-성질" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>성질</span> </div> </a> <button aria-controls="toc-성질-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>성질 하위섹션 토글하기</span> </button> <ul id="toc-성질-sublist" class="vector-toc-list"> <li id="toc-완비화" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#완비화"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>완비화</span> </div> </a> <ul id="toc-완비화-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-하이네-보렐_정리" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#하이네-보렐_정리"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>하이네-보렐 정리</span> </div> </a> <ul id="toc-하이네-보렐_정리-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-베르_범주_정리" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#베르_범주_정리"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>베르 범주 정리</span> </div> </a> <ul id="toc-베르_범주_정리-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-바나흐_고정점_정리" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#바나흐_고정점_정리"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>바나흐 고정점 정리</span> </div> </a> <ul id="toc-바나흐_고정점_정리-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-예" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#예"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>예</span> </div> </a> <button aria-controls="toc-예-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>예 하위섹션 토글하기</span> </button> <ul id="toc-예-sublist" class="vector-toc-list"> <li id="toc-실직선_속의_코시_수열" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#실직선_속의_코시_수열"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>실직선 속의 코시 수열</span> </div> </a> <ul id="toc-실직선_속의_코시_수열-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-이산_공간" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#이산_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>이산 공간</span> </div> </a> <ul id="toc-이산_공간-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-완비_공간_값의_유계_함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#완비_공간_값의_유계_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>완비 공간 값의 유계 함수</span> </div> </a> <ul id="toc-완비_공간_값의_유계_함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-바나흐_공간" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#바나흐_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>바나흐 공간</span> </div> </a> <ul id="toc-바나흐_공간-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-역사" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>역사</span> </div> </a> <ul id="toc-역사-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="목차" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="목차 토글" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">목차 토글</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">완비 거리 공간</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="다른 언어로 문서를 방문합니다. 30개 언어로 읽을 수 있습니다" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-30" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">30개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D9%83%D8%A7%D9%85%D9%84" title="فضاء كامل – 아랍어" lang="ar" hreflang="ar" data-title="فضاء كامل" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_complet" title="Espai complet – 카탈로니아어" lang="ca" hreflang="ca" data-title="Espai complet" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C3%9Apln%C3%BD_metrick%C3%BD_prostor" title="Úplný metrický prostor – 체코어" lang="cs" hreflang="cs" data-title="Úplný metrický prostor" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fuldst%C3%A6ndigt_metrisk_rum" title="Fuldstændigt metrisk rum – 덴마크어" lang="da" hreflang="da" data-title="Fuldstændigt metrisk rum" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vollst%C3%A4ndiger_Raum" title="Vollständiger Raum – 독일어" lang="de" hreflang="de" data-title="Vollständiger Raum" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BB%CE%AE%CF%81%CE%B7%CF%82_%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Πλήρης μετρικός χώρος – 그리스어" lang="el" hreflang="el" data-title="Πλήρης μετρικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Complete_metric_space" title="Complete metric space – 영어" lang="en" hreflang="en" data-title="Complete metric space" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompleta_metrika_spaco" title="Kompleta metrika spaco – 에스페란토어" lang="eo" hreflang="eo" data-title="Kompleta metrika spaco" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_m%C3%A9trico_completo" title="Espacio métrico completo – 스페인어" lang="es" hreflang="es" data-title="Espacio métrico completo" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%85%D8%AA%D8%B1%DB%8C%DA%A9_%DA%A9%D8%A7%D9%85%D9%84" title="فضای متریک کامل – 페르시아어" lang="fa" hreflang="fa" data-title="فضای متریک کامل" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/T%C3%A4ydellisyys" title="Täydellisyys – 핀란드어" lang="fi" hreflang="fi" data-title="Täydellisyys" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_complet" title="Espace complet – 프랑스어" lang="fr" hreflang="fr" data-title="Espace complet" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_completo" title="Espazo completo – 갈리시아어" lang="gl" hreflang="gl" data-title="Espazo completo" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%9E%D7%98%D7%A8%D7%99_%D7%A9%D7%9C%D7%9D" title="מרחב מטרי שלם – 히브리어" lang="he" hreflang="he" data-title="מרחב מטרי שלם" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BC%D6%80%D5%AB%D5%BE_%D5%B4%D5%A5%D5%BF%D6%80%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Լրիվ մետրիկական տարածություն – 아르메니아어" lang="hy" hreflang="hy" data-title="Լրիվ մետրիկական տարածություն" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_metrik_lengkap" title="Ruang metrik lengkap – 인도네시아어" lang="id" hreflang="id" data-title="Ruang metrik lengkap" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fullkomi%C3%B0_fir%C3%B0r%C3%BAm" title="Fullkomið firðrúm – 아이슬란드어" lang="is" hreflang="is" data-title="Fullkomið firðrúm" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_metrico_completo" title="Spazio metrico completo – 이탈리아어" lang="it" hreflang="it" data-title="Spazio metrico completo" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AE%8C%E5%82%99%E8%B7%9D%E9%9B%A2%E7%A9%BA%E9%96%93" title="完備距離空間 – 일본어" lang="ja" hreflang="ja" data-title="完備距離空間" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a 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</p> <div class="dablink hatnote"><span typeof="mw:File"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%8F%99%EC%9D%8C%EC%9D%B4%EC%9D%98%EC%96%B4_%EB%AC%B8%EC%84%9C" title="위키백과:동음이의어 문서"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/23px-Disambig_grey.svg.png" decoding="async" width="23" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/35px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/46px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span> <b>완비화</b>는 여기로 연결됩니다. 꽃받침, 꽃잎, 암술과 수술을 모두 갖춘 꽃에 대해서는 <a href="/w/index.php?title=%EA%B0%96%EC%B6%98%EA%BD%83&action=edit&redlink=1" class="new" title="갖춘꽃 (없는 문서)">갖춘꽃</a> 문서를, 환론에서의 연산에 대해서는 <a href="/wiki/%EC%99%84%EB%B9%84%ED%99%94_(%ED%99%98%EB%A1%A0)" title="완비화 (환론)">완비화 (환론)</a> 문서를 참고하십시오.</div> <p><a href="/wiki/%EA%B8%B0%ED%95%98%ED%95%99" title="기하학">기하학</a>에서 <b>완비 거리 공간</b>(完備距離空間, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">complete metric space</span>)은 그 안이나 경계에 "빠진 점"이 없는 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>이다. 완비 거리 공간의 정의는 <b>코시 열</b>(Cauchy列, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Cauchy sequence</span>)이라는 개념을 사용한다. 코시 열은 점들 사이의 거리가 서로 점점 가까워지는 <a href="/wiki/%EC%88%98%EC%97%B4" title="수열">수열</a>이다. 즉, 코시 열에서는 충분한 수의 처음 유한 개의 점들을 제외하면, 남은 점들 사이의 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리</a>가 임의로 작아진다. 완비 거리 공간은 이렇게 "수렴하는 것처럼 보이는" 점렬들이 모두 실제로 수렴하는 점을 갖는 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>이다. <a href="/wiki/%EC%99%84%EB%B9%84_%EA%B7%A0%EB%93%B1_%EA%B3%B5%EA%B0%84" title="완비 균등 공간">완비 균등 공간</a>의 특수한 경우이다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="코시_열"><span id=".EC.BD.94.EC.8B.9C_.EC.97.B4"></span>코시 열</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=2" title="부분 편집: 코시 열"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%BD%94%EC%8B%9C_%EC%97%B4" title="코시 열">코시 열</a>입니다.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Cauchy_sequence_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/220px-Cauchy_sequence_illustration.svg.png" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/330px-Cauchy_sequence_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/440px-Cauchy_sequence_illustration.svg.png 2x" data-file-width="305" data-file-height="170" /></a><figcaption>코시 열의 예. 코시 열에서는 점 사이의 거리가 0으로 수렴한다.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Cauchy_sequence_illustration2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/220px-Cauchy_sequence_illustration2.svg.png" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/330px-Cauchy_sequence_illustration2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/440px-Cauchy_sequence_illustration2.svg.png 2x" data-file-width="305" data-file-height="170" /></a><figcaption>코시 열이 아닌 수열</figcaption></figure> <p><a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span> 위의 <a href="/wiki/%EC%88%98%EC%97%B4" title="수열">수열</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{i=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})_{i=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8143b6d5acd698e4cb1ede5e73f58b89dfa4f572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.839ex; height:3.009ex;" alt="{\displaystyle (x_{i})_{i=0}^{\infty }}"></span>이 있다고 하자. 만약 임의의 양의 실수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>에 대하여, 다음 조건을 만족시키는 <a href="/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수">자연수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\epsilon )\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\epsilon )\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616334231c5269acffa23d9df6d3f213884f7ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.336ex; height:2.843ex;" alt="{\displaystyle N(\epsilon )\in \mathbb {N} }"></span>가 존재한다고 하자. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x_{i},x_{j})<\epsilon \qquad \forall i,j\geq N(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi>ϵ</mi> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≥</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x_{i},x_{j})<\epsilon \qquad \forall i,j\geq N(\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc9c8474474c36af8b7aac1e46f694212d9c1a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.118ex; height:3.009ex;" alt="{\displaystyle d(x_{i},x_{j})<\epsilon \qquad \forall i,j\geq N(\epsilon )}"></span></dd></dl> <p>그렇다면 수열 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{i=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})_{i=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8143b6d5acd698e4cb1ede5e73f58b89dfa4f572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.839ex; height:3.009ex;" alt="{\displaystyle (x_{i})_{i=0}^{\infty }}"></span>를 <b>코시 열</b>이라고 한다. 임의의 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> 속에서, 모든 수렴하는 수열은 코시 열을 이루며, 모든 코시 열은 <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%A7%91%ED%95%A9" title="유계 집합">유계 집합</a>을 이룬다. 즉, 다음과 같은 포함 관계가 성립한다. </p> <dl><dd><a href="/wiki/%EC%83%81%EC%88%98_%ED%95%A8%EC%88%98" title="상수 함수">상수 점렬</a> ⊆ 수렴 점렬 ⊆ 코시 점렬 ⊆ <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%A7%91%ED%95%A9" title="유계 집합">유계 점렬</a></dd></dl> <div class="mw-heading mw-heading3"><h3 id="확대_상수"><span id=".ED.99.95.EB.8C.80_.EC.83.81.EC.88.98"></span>확대 상수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=3" title="부분 편집: 확대 상수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span> 및 실수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a01385c923ba1204024e8c71b06488073ec04fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.314ex; height:2.843ex;" alt="{\displaystyle \mu \in [0,\infty )}"></span>에 대하여, 다음 조건이 성립하는지 여부를 물을 수 있다. </p> <ul><li>임의의 점들의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cbb47cf9bb3374016df9c9c71f54f5b28ff475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.663ex; height:2.843ex;" alt="{\displaystyle (x_{i})_{i\in I}}"></span> 및 반지름들의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e67682c5285672cf8deac703c5cd55d378a22d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.382ex; height:2.843ex;" alt="{\displaystyle (r_{i})_{i\in I}}"></span>에 대하여, 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall i,j\in I\colon {\bar {B}}(x_{i},r_{i})\cap {\bar {B}}(x_{j},r_{j})\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mi>I</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall i,j\in I\colon {\bar {B}}(x_{i},r_{i})\cap {\bar {B}}(x_{j},r_{j})\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3af17ef6043688ba7cf0b11e4e22fdc4e45b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.012ex; height:3.176ex;" alt="{\displaystyle \forall i,j\in I\colon {\bar {B}}(x_{i},r_{i})\cap {\bar {B}}(x_{j},r_{j})\neq \varnothing }"></span>이라면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigcap _{i\in I}{\bar {B}}(x_{i},\mu r_{i})\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>μ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \bigcap _{i\in I}{\bar {B}}(x_{i},\mu r_{i})\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/140119a512fc9be1c3a0c86a6869bd579b503ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.941ex; height:3.176ex;" alt="{\displaystyle \textstyle \bigcap _{i\in I}{\bar {B}}(x_{i},\mu r_{i})\neq \varnothing }"></span>이다.</li></ul> <p>거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span>의 <b>확대 상수</b>(擴大常數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">expansion constant</span>) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(X)\in [0,\infty ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X)\in [0,\infty ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70920cb989ea80e44f1e1a656aac2bf4713bb187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.219ex; height:2.843ex;" alt="{\displaystyle E(X)\in [0,\infty ]}"></span></dd></dl> <p>는 위 조건을 만족시키는 모든 실수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>들의 <a href="/wiki/%ED%95%98%ED%95%9C" class="mw-redirect" title="하한">하한</a>이다. </p><p>임의의 거리 공간에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(X)\in \{\infty \}\cup [0,2]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X)\in \{\infty \}\cup [0,2]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6e3e9aed931f1cc7ad3f8dcc2f637be8bf4463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.289ex; height:2.843ex;" alt="{\displaystyle E(X)\in \{\infty \}\cup [0,2]}"></span>이다.<sup id="cite_ref-Grunbaum_1-0" class="reference"><a href="#cite_note-Grunbaum-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:194, 198</sup></span> </p> <div class="mw-heading mw-heading3"><h3 id="카리스티_사상과_칸난_사상"><span id=".EC.B9.B4.EB.A6.AC.EC.8A.A4.ED.8B.B0_.EC.82.AC.EC.83.81.EA.B3.BC_.EC.B9.B8.EB.82.9C_.EC.82.AC.EC.83.81"></span>카리스티 사상과 칸난 사상</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=4" title="부분 편집: 카리스티 사상과 칸난 사상"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span> 위의 <b>카리스티 사상</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Caristi map</span>)은 다음 조건을 만족시키는 <a href="/wiki/%ED%95%98%EB%B0%98%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" class="mw-redirect" title="하반연속 함수">하반연속 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon X\to [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \colon X\to [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af968874efa9b5b1dc3b828b2adce07f293efa16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.085ex; height:2.843ex;" alt="{\displaystyle \phi \colon X\to [0,\infty )}"></span>가 존재하는 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773c98269d8a3c82725e5cb650243666484b528b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.887ex; height:2.509ex;" alt="{\displaystyle f\colon X\to X}"></span>이다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,f(x))\leq \phi (x)-\phi (f(x))\qquad \forall x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,f(x))\leq \phi (x)-\phi (f(x))\qquad \forall x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74140c885847135f6463457dc6d38f7a29864b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.97ex; height:2.843ex;" alt="{\displaystyle d(x,f(x))\leq \phi (x)-\phi (f(x))\qquad \forall x\in X}"></span></dd></dl> <p><a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span> 위의 <b>칸난 사상</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Kannan map</span>)은 다음 조건을 만족시키는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\in [0,1/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\in [0,1/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec9028da122c1c6b57e98f7c39fa833367752c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.842ex; height:2.843ex;" alt="{\displaystyle C\in [0,1/2)}"></span>가 존재하는 함수 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773c98269d8a3c82725e5cb650243666484b528b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.887ex; height:2.509ex;" alt="{\displaystyle f\colon X\to X}"></span>이다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f(x),f(y))\leq C(d(x,f(x))+d(y,f(y)))\qquad \forall x,y\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f(x),f(y))\leq C(d(x,f(x))+d(y,f(y)))\qquad \forall x,y\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93a55b24b40cb935705e48a9bd8515a72795beac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.776ex; height:2.843ex;" alt="{\displaystyle d(f(x),f(y))\leq C(d(x,f(x))+d(y,f(y)))\qquad \forall x,y\in X}"></span></dd></dl> <p>모든 칸난 사상은 카리스티 사상이다. </p> <div class="mw-heading mw-heading3"><h3 id="완비_거리_공간"><span id=".EC.99.84.EB.B9.84_.EA.B1.B0.EB.A6.AC_.EA.B3.B5.EA.B0.84"></span>완비 거리 공간</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=5" title="부분 편집: 완비 거리 공간"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span>에 대하여 다음 조건들이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이며, 이를 만족시키는 거리 공간을 <b>완비 거리 공간</b>이라고 한다. </p> <ul><li>모든 코시 점렬이 수렴한다.</li> <li>확대 상수가 유한하다.<sup id="cite_ref-Grunbaum_1-1" class="reference"><a href="#cite_note-Grunbaum-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:194, 198</sup></span></li> <li>확대 상수가 2 이하이다.<sup id="cite_ref-Grunbaum_1-2" class="reference"><a href="#cite_note-Grunbaum-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:194, 198</sup></span></li> <li>임의의 <a href="/wiki/%EB%8B%AB%ED%9E%8C_%EA%B3%B5" class="mw-redirect" title="닫힌 공">닫힌 공</a>들의 감소열 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\supseteq {\bar {B}}(x_{0},r_{0})\supseteq {\bar {B}}(x_{1},r_{1})\supseteq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊇</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊇</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊇</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\supseteq {\bar {B}}(x_{0},r_{0})\supseteq {\bar {B}}(x_{1},r_{1})\supseteq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8ddcc5bd4925a618db48c1187572b527fe97d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.187ex; height:3.009ex;" alt="{\displaystyle X\supseteq {\bar {B}}(x_{0},r_{0})\supseteq {\bar {B}}(x_{1},r_{1})\supseteq \cdots }"></span>이 주어졌고, 그 반지름들이 0으로 수렴하며 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lim _{i\to \infty }r_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \lim _{i\to \infty }r_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be36fd443dfe59bdf2677fa79fd675eb4f83cb17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.812ex; height:2.509ex;" alt="{\displaystyle \textstyle \lim _{i\to \infty }r_{i}=0}"></span>), 모두 <a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a>이 아니라고 하자 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}>0\;\forall i\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}>0\;\forall i\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64fa9cb110f2a3c7191c5af3ba9c0dfa01d2dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.368ex; height:2.509ex;" alt="{\displaystyle r_{i}>0\;\forall i\in \mathbb {N} }"></span>). 그렇다면 이들의 <a href="/wiki/%EA%B5%90%EC%A7%91%ED%95%A9" title="교집합">교집합</a>은 <a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a>이 아니다 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigcap _{i=0}^{\infty }{\bar {B}}(x_{i},r_{i})\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \bigcap _{i=0}^{\infty }{\bar {B}}(x_{i},r_{i})\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab480cc8b9f92b6410d819b816aef6003832229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.715ex; height:3.176ex;" alt="{\displaystyle \textstyle \bigcap _{i=0}^{\infty }{\bar {B}}(x_{i},r_{i})\neq \varnothing }"></span>).</li> <li>임의의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>들의 감소열 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\supseteq C_{0}\supseteq C_{1}\supseteq C_{2}\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊇</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊇</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊇</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\supseteq C_{0}\supseteq C_{1}\supseteq C_{2}\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4955c4e9e87a21f70696a3f2651be610467752f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.534ex; height:2.509ex;" alt="{\displaystyle X\supseteq C_{0}\supseteq C_{1}\supseteq C_{2}\cdots }"></span>이 주어졌고, 그 <a href="/wiki/%EC%A7%80%EB%A6%84" title="지름">지름</a>들이 0으로 수렴하며 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lim _{i\to \infty }\operatorname {diam} C_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>diam</mi> <mo>⁡</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \lim _{i\to \infty }\operatorname {diam} C_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b9cb03c36a55ac6b7de2f0f47ab6cc60eaede2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.85ex; height:2.509ex;" alt="{\displaystyle \textstyle \lim _{i\to \infty }\operatorname {diam} C_{i}=0}"></span>), 모두 <a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a>이 아니라고 하자 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}\neq \varnothing \;\forall i\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}\neq \varnothing \;\forall i\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0748431f66f6c72d070f61055dbcaf516f18ca68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.627ex; height:2.676ex;" alt="{\displaystyle C_{i}\neq \varnothing \;\forall i\in \mathbb {N} }"></span>). 그렇다면 이들의 <a href="/wiki/%EA%B5%90%EC%A7%91%ED%95%A9" title="교집합">교집합</a>은 <a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a>이 아니다 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigcap _{i=0}^{\infty }C_{i}\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \bigcap _{i=0}^{\infty }C_{i}\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa62be81b947e3db4460e0a836e8d8ba985aec57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.591ex; height:3.176ex;" alt="{\displaystyle \textstyle \bigcap _{i=0}^{\infty }C_{i}\neq \varnothing }"></span>).</li> <li>(<b>카리스티 고정점 정리</b>, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Caristi fixed-point theorem</span>) 모든 카리스티 사상은 <a href="/wiki/%EA%B3%A0%EC%A0%95%EC%A0%90" title="고정점">고정점</a>을 갖는다.</li> <li>(<b>칸난 고정점 정리</b>, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Kannan fixed-point theorem</span>) 모든 칸난 사상은 <a href="/wiki/%EA%B3%A0%EC%A0%95%EC%A0%90" title="고정점">고정점</a>을 갖는다.</li></ul> <div class="mw-heading mw-heading2"><h2 id="성질"><span id=".EC.84.B1.EC.A7.88"></span>성질</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=6" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>완비 거리 공간 속의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>은 완비 거리 공간을 이룬다. 반대로, <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>의 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</a>이 완비 거리 공간을 이룬다면, 이는 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>이다. </p> <div class="mw-heading mw-heading3"><h3 id="완비화"><span id=".EC.99.84.EB.B9.84.ED.99.94"></span>완비화</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=7" title="부분 편집: 완비화"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span>의 <b>완비화</b>(完備化, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">completion</span>)는 다음과 같다. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>의 모든 코시 점렬의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cauchy} (X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cauchy</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cauchy} (X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c6551168cf4b56a1875a484c26c3c6eeadcf9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.475ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cauchy} (X)}"></span>에 다음과 같은 <a href="/wiki/%EC%9C%A0%EC%82%AC_%EA%B1%B0%EB%A6%AC_%ED%95%A8%EC%88%98" class="mw-redirect" title="유사 거리 함수">유사 거리 함수</a>를 주자. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d((x_{i})_{i\in \mathbb {N} },(y_{i})_{i\in \mathbb {N} })=\lim _{i\to \infty }d(x_{i},y_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d((x_{i})_{i\in \mathbb {N} },(y_{i})_{i\in \mathbb {N} })=\lim _{i\to \infty }d(x_{i},y_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2e960b999f3aacec6d168f619fcca4424e73f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.378ex; height:4.009ex;" alt="{\displaystyle d((x_{i})_{i\in \mathbb {N} },(y_{i})_{i\in \mathbb {N} })=\lim _{i\to \infty }d(x_{i},y_{i})}"></span></dd></dl> <p>코시 점렬의 정의에 따라 이 극한은 항상 존재한다. 이를 부여하면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cauchy} (X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cauchy</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cauchy} (X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c6551168cf4b56a1875a484c26c3c6eeadcf9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.475ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cauchy} (X)}"></span>는 <a href="/wiki/%EC%9C%A0%EC%82%AC_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="유사 거리 공간">유사 거리 공간</a>을 이루지만, 거리가 0인 서로 다른 코시 점렬이 존재하므로 거리 공간이 아니다. 이 경우, 거리가 0인 코시 점렬들을 서로 동치로 간주하는 <a href="/wiki/%EB%8F%99%EC%B9%98_%EA%B4%80%EA%B3%84" title="동치 관계">동치 관계</a>를 정의하자. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{i\in \mathbb {N} }\sim (y_{i})_{i\in \mathbb {N} }\iff \lim _{i\to \infty }d(x_{i},y_{i})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>∼</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})_{i\in \mathbb {N} }\sim (y_{i})_{i\in \mathbb {N} }\iff \lim _{i\to \infty }d(x_{i},y_{i})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa168a01da743522f2b3d2db80f557fb182998dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.477ex; height:4.009ex;" alt="{\displaystyle (x_{i})_{i\in \mathbb {N} }\sim (y_{i})_{i\in \mathbb {N} }\iff \lim _{i\to \infty }d(x_{i},y_{i})=0}"></span></dd></dl> <p>즉, 무한히 가까워지는 두 코시 점렬들을 같은 <a href="/wiki/%EB%8F%99%EC%B9%98%EB%A5%98" class="mw-redirect" title="동치류">동치류</a>에 넣는다. 이렇게 하면, <a href="/wiki/%EB%AA%AB%EC%A7%91%ED%95%A9" class="mw-redirect" title="몫집합">몫집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cauchy} (X)/{\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cauchy</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cauchy} (X)/{\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a57ab075cc554a43feed041ce7aa578b81d9732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.445ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cauchy} (X)/{\sim }}"></span> 위에 거리가 유일하게 정의되며, 이는 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>을 이루며 또한 완비 거리 공간이 된다. 이를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>의 <b>완비화</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}"></span>라고 한다. </p><p><a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>에서 그 완비화 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}"></span>로 가는 표준적인 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\hookrightarrow {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↪</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\hookrightarrow {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c479ee9af604e82500199b671713f01ab489c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.867ex; height:2.509ex;" alt="{\displaystyle X\hookrightarrow {\bar {X}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto [(x,x,x,\ldots )]_{\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto [(x,x,x,\ldots )]_{\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c32b3301684af9f113d758bc8ac704d2b77068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.372ex; height:2.843ex;" alt="{\displaystyle x\mapsto [(x,x,x,\ldots )]_{\sim }}"></span></dd></dl> <p>가 존재한다. 이 함수는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>의 각 점을 (자명하게 코시 점렬을 이루는) <a href="/wiki/%EC%83%81%EC%88%98_%ED%95%A8%EC%88%98" title="상수 함수">상수 점렬</a>의 <a href="/wiki/%EB%8F%99%EC%B9%98%EB%A5%98" class="mw-redirect" title="동치류">동치류</a>로 대응시킨다. 이는 <a href="/wiki/%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="단사 함수">단사</a> <a href="/wiki/%EB%93%B1%EA%B1%B0%EB%A6%AC%EB%B3%80%ED%99%98" title="등거리변환">등거리변환</a>이며, 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>가 완비 거리 공간이라면 이는 거리 공간의 동형이다. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}"></span>의 부분 집합으로서, <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%A7%91%ED%95%A9" title="조밀 집합">조밀 집합</a>이다. </p> <style data-mw-deduplicate="TemplateStyles:r26858958">.mw-parser-output div.proof{border:1px solid #aaaaaa;background-color:#f9f9f9;padding:5px;font-size:95%;min-width:50%}.mw-parser-output div.proof,.mw-parser-output div.prooftitle,.mw-parser-output div.proofcontent{overflow:auto}.mw-parser-output div.prooftitle span.prooftitletext{font-weight:bold}.mw-parser-output div.proofcontent{margin-top:-0.5em;min-height:0.5em}</style><div class="proof mw-collapsible mw-collapsed"> <div class="prooftitle"> <p><span class="prooftitletext">증명 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>의 조밀성):</span> </p> </div> <div class="proofcontent mw-collapsible-content"> <p>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(x_{i})_{i\in \mathbb {N} }]_{\sim }\in {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(x_{i})_{i\in \mathbb {N} }]_{\sim }\in {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60e80c022dc1072f4710c635f2303a67238c3d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.646ex; height:3.009ex;" alt="{\displaystyle [(x_{i})_{i\in \mathbb {N} }]_{\sim }\in {\bar {X}}}"></span>에 대하여, 점렬 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ([(x_{i},x_{i},x_{i},\ldots )]_{\sim })_{i\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ([(x_{i},x_{i},x_{i},\ldots )]_{\sim })_{i\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeea0eb3605eb0a5b7c31ce55a3a10cb51738c8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.719ex; height:2.843ex;" alt="{\displaystyle ([(x_{i},x_{i},x_{i},\ldots )]_{\sim })_{i\in \mathbb {N} }}"></span></dd></dl> <p>은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{i\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})_{i\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b467539a8a95273842a51d4f6b392b2a0568fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.021ex; height:2.843ex;" alt="{\displaystyle (x_{i})_{i\in \mathbb {N} }}"></span>으로 수렴한다. </p> </div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26858958"><div class="proof mw-collapsible mw-collapsed"> <div class="prooftitle"> <p><span class="prooftitletext">증명 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b968141b314f4de17f5e63f18dcdc126352bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.509ex;" alt="{\displaystyle {\bar {X}}}"></span>의 완비성):</span> </p> </div> <div class="proofcontent mw-collapsible-content"> <p>임의의 코시 점렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ([(x_{i,j})_{j\in \mathbb {N} }]_{\sim })_{i\in \mathbb {N} }\subseteq {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ([(x_{i,j})_{j\in \mathbb {N} }]_{\sim })_{i\in \mathbb {N} }\subseteq {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c930c035d5ded07b5ba1c88806974c0f827ff7ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.041ex; height:3.176ex;" alt="{\displaystyle ([(x_{i,j})_{j\in \mathbb {N} }]_{\sim })_{i\in \mathbb {N} }\subseteq {\bar {X}}}"></span>가 주어졌다고 하자. 그렇다면, 각 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e64c8c5906eb3eb9d7a8b1ed1e31de4e5fc6c632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.321ex; height:2.176ex;" alt="{\displaystyle i\in \mathbb {N} }"></span>에 대하여, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d([(x_{i,j})_{j\in \mathbb {N} }]_{\sim },[(y_{i},y_{i},y_{i},\ldots )]_{\sim })<1/i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <mo>,</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d([(x_{i,j})_{j\in \mathbb {N} }]_{\sim },[(y_{i},y_{i},y_{i},\ldots )]_{\sim })<1/i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c274b3910dda4f150b14532235e04bdd0244ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.611ex; height:3.009ex;" alt="{\displaystyle d([(x_{i,j})_{j\in \mathbb {N} }]_{\sim },[(y_{i},y_{i},y_{i},\ldots )]_{\sim })<1/i}"></span></dd></dl> <p>인 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4297077e7998ea26923a9cf4e7fe7694fa2e58a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.76ex; height:2.509ex;" alt="{\displaystyle y_{i}\in X}"></span>가 존재한다. 이 경우, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{i})_{i\in \mathbb {N} }\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y_{i})_{i\in \mathbb {N} }\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41726a34c6beed54ccb471f96a61356167846954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.909ex; height:2.843ex;" alt="{\displaystyle (y_{i})_{i\in \mathbb {N} }\subseteq X}"></span>는 코시 점렬이며, 점렬 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ([(y_{i},y_{i},y_{i},\ldots )]_{\sim })_{i\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ([(y_{i},y_{i},y_{i},\ldots )]_{\sim })_{i\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c5e247f2d0f211b439ebd6aa12b5e9a067cef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.147ex; height:2.843ex;" alt="{\displaystyle ([(y_{i},y_{i},y_{i},\ldots )]_{\sim })_{i\in \mathbb {N} }}"></span></dd></dl> <p>은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(y_{i})_{i\in \mathbb {N} }]_{\sim }\in {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(y_{i})_{i\in \mathbb {N} }]_{\sim }\in {\bar {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed9f352cf51ebfb86f54e33ea659ebdada29306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.456ex; height:3.009ex;" alt="{\displaystyle [(y_{i})_{i\in \mathbb {N} }]_{\sim }\in {\bar {X}}}"></span>으로 수렴한다. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ([(x_{i,j})_{j\in \mathbb {N} }]_{\sim })_{i\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ([(x_{i,j})_{j\in \mathbb {N} }]_{\sim })_{i\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac90e8aa22958e9223582a2e4ffb567c85ede62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.962ex; height:3.009ex;" alt="{\displaystyle ([(x_{i,j})_{j\in \mathbb {N} }]_{\sim })_{i\in \mathbb {N} }}"></span>은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [(y_{i})_{i\in \mathbb {N} }]_{\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(y_{i})_{i\in \mathbb {N} }]_{\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04466f28ec1b0f5917bf1af3898574c825db118d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.635ex; height:2.843ex;" alt="{\displaystyle [(y_{i})_{i\in \mathbb {N} }]_{\sim }}"></span>으로 수렴한다. </p> </div></div> <div class="mw-heading mw-heading3"><h3 id="하이네-보렐_정리"><span id=".ED.95.98.EC.9D.B4.EB.84.A4-.EB.B3.B4.EB.A0.90_.EC.A0.95.EB.A6.AC"></span>하이네-보렐 정리</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=8" title="부분 편집: 하이네-보렐 정리"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%ED%95%98%EC%9D%B4%EB%84%A4-%EB%B3%B4%EB%A0%90_%EC%A0%95%EB%A6%AC" title="하이네-보렐 정리">하이네-보렐 정리</a>입니다.</div> <p>모든 <a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간">콤팩트</a> <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>은 완비 거리 공간이다. 사실, <a href="/wiki/%ED%95%98%EC%9D%B4%EB%84%A4-%EB%B3%B4%EB%A0%90_%EC%A0%95%EB%A6%AC" title="하이네-보렐 정리">하이네-보렐 정리</a>에 따르면, <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>에 대하여, 콤팩트 공간인 것은 완비 <a href="/wiki/%EC%99%84%EC%A0%84_%EC%9C%A0%EA%B3%84_%EA%B3%B5%EA%B0%84" title="완전 유계 공간">완전 유계 공간</a>인 것과 동치이다. </p> <div class="mw-heading mw-heading3"><h3 id="베르_범주_정리"><span id=".EB.B2.A0.EB.A5.B4_.EB.B2.94.EC.A3.BC_.EC.A0.95.EB.A6.AC"></span>베르 범주 정리</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=9" title="부분 편집: 베르 범주 정리"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%B2%A0%EB%A5%B4_%EB%B2%94%EC%A3%BC_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="베르 범주 정리">베르 범주 정리</a>입니다.</div> <p><a href="/wiki/%EB%B2%A0%EB%A5%B4_%EB%B2%94%EC%A3%BC_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="베르 범주 정리">베르 범주 정리</a>에 따르면, 모든 완비 거리 공간은 <a href="/wiki/%EB%B2%A0%EB%A5%B4_%EA%B3%B5%EA%B0%84" title="베르 공간">베르 공간</a>이다. </p> <div class="mw-heading mw-heading3"><h3 id="바나흐_고정점_정리"><span id=".EB.B0.94.EB.82.98.ED.9D.90_.EA.B3.A0.EC.A0.95.EC.A0.90_.EC.A0.95.EB.A6.AC"></span>바나흐 고정점 정리</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=10" title="부분 편집: 바나흐 고정점 정리"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%B0%94%EB%82%98%ED%9D%90_%EA%B3%A0%EC%A0%95%EC%A0%90_%EC%A0%95%EB%A6%AC" title="바나흐 고정점 정리">바나흐 고정점 정리</a>입니다.</div> <p><a href="/wiki/%EB%B0%94%EB%82%98%ED%9D%90_%EA%B3%A0%EC%A0%95%EC%A0%90_%EC%A0%95%EB%A6%AC" title="바나흐 고정점 정리">바나흐 고정점 정리</a>에 따르면, 완비 거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 위의 <a href="/wiki/%EC%B6%95%EC%95%BD_%EC%82%AC%EC%83%81" class="mw-redirect" title="축약 사상">축약 사상</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773c98269d8a3c82725e5cb650243666484b528b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.887ex; height:2.509ex;" alt="{\displaystyle f\colon X\to X}"></span>은 유일한 <a href="/wiki/%EA%B3%A0%EC%A0%95%EC%A0%90" title="고정점">고정점</a>을 갖는다. 모든 <a href="/wiki/%EC%B6%95%EC%95%BD_%EC%82%AC%EC%83%81" class="mw-redirect" title="축약 사상">축약 사상</a>은 카리스티 사상이므로, 이는 카리스티 고정점 정리의 특수한 경우이다. 하지만 바나흐 고정점 정리는 완비 거리 공간의 정의로 삼을 수 없다. </p> <div class="mw-heading mw-heading2"><h2 id="예"><span id=".EC.98.88"></span>예</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=11" title="부분 편집: 예"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="실직선_속의_코시_수열"><span id=".EC.8B.A4.EC.A7.81.EC.84.A0_.EC.86.8D.EC.9D.98_.EC.BD.94.EC.8B.9C_.EC.88.98.EC.97.B4"></span>실직선 속의 코시 수열</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=12" title="부분 편집: 실직선 속의 코시 수열"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a> 전체의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>와 <a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a> 전체의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>에 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92" title="절댓값">절댓값</a>으로 정의되는 일반적인 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%ED%95%A8%EC%88%98" title="거리 함수">거리 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>로 정의된 거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} ,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} ,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd797a0e3370f170adb8cfd4c2dfe574a3ae0384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.867ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} ,d)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b9dc9e0b336f8316bf4b916e1ad2e4786eadcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.737ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,d)}"></span>가 있을 때, 수열 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1/n\}_{n\in \mathbf {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1/n\}_{n\in \mathbf {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fca92eb8d3cd0ccff23b0180fe50be449b6501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.838ex; height:2.843ex;" alt="{\displaystyle \{1/n\}_{n\in \mathbf {N} }}"></span>은 코시 수열이다. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> 모두에서 수렴하며, 수렴하는 값은 0이다. </p><p><a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a>의 거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} ,|\cdot |)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} ,|\cdot |)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d11e872b6a21d4f1175ac81271f294503aaf3b2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.624ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} ,|\cdot |)}"></span>는 완비 거리 공간이 아닌데, 이는 그 안에서 <a href="/wiki/%EB%AC%B4%EB%A6%AC%EC%88%98" title="무리수">무리수</a>인 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>로 가까워지는 코시 수열을 만들 수 있기 때문이다. 구체적으로, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=\lfloor n{\sqrt {2}}\rfloor /n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo fence="false" stretchy="false">⌋</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}=\lfloor n{\sqrt {2}}\rfloor /n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3d3c0e898eacf02bdc0867b3b1075818d05274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.762ex; height:3.176ex;" alt="{\displaystyle x_{n}=\lfloor n{\sqrt {2}}\rfloor /n}"></span> 로 정의된 수열 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810ff85512c8477ba6aff18f4480ffeeac0bb3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.374ex; height:2.843ex;" alt="{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}"></span>은 코시 수열이다. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>에서는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>로 수렴하지만, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>는 유리수가 아니므로 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>에서는 수렴하지 않는다. 유리수 공간의 완비화는 실수의 거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,|\cdot |)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,|\cdot |)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8617fa084440d7e0966d3cc400de98c2d02f0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.494ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,|\cdot |)}"></span>이다. </p> <div class="mw-heading mw-heading3"><h3 id="이산_공간"><span id=".EC.9D.B4.EC.82.B0_.EA.B3.B5.EA.B0.84"></span>이산 공간</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=13" title="부분 편집: 이산 공간"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%9D%B4%EC%82%B0_%EA%B3%B5%EA%B0%84" title="이산 공간">이산 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 위에 이산 거리 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)={\begin{cases}1&x\neq y\\0&x=y\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)={\begin{cases}1&x\neq y\\0&x=y\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3deea749ea8477d1bf6425fd15beabc0f9d55c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.206ex; height:6.176ex;" alt="{\displaystyle d(x,y)={\begin{cases}1&x\neq y\\0&x=y\end{cases}}}"></span></dd></dl> <p>를 준다면, 그 속의 점렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{i=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})_{i=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8143b6d5acd698e4cb1ede5e73f58b89dfa4f572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.839ex; height:3.009ex;" alt="{\displaystyle (x_{i})_{i=0}^{\infty }}"></span>에 대하여 다음 세 조건이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li>결국 상수 점렬이다. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{N}=x_{N+1}=x_{N+2}=\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{N}=x_{N+1}=x_{N+2}=\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f40f457f15710a9a0daa1628c4872ff8c65076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.283ex; height:2.009ex;" alt="{\displaystyle x_{N}=x_{N+1}=x_{N+2}=\cdots }"></span>가 되는 자연수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b985ba501f78cb9890f3ecda3e2e315cbd5cb26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.582ex; height:2.176ex;" alt="{\displaystyle N\in \mathbb {N} }"></span>이 존재한다.</li> <li>수렴 점렬이다.</li> <li>코시 점렬이다.</li></ul> <p>따라서 이산 공간은 완비 거리 공간을 이룬다. </p> <div class="mw-heading mw-heading3"><h3 id="완비_공간_값의_유계_함수"><span id=".EC.99.84.EB.B9.84_.EA.B3.B5.EA.B0.84_.EA.B0.92.EC.9D.98_.EC.9C.A0.EA.B3.84_.ED.95.A8.EC.88.98"></span>완비 공간 값의 유계 함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=14" title="부분 편집: 완비 공간 값의 유계 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>임의의 <a href="/wiki/%EC%A7%91%ED%95%A9" title="집합">집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> 및 완비 거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(S,X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(S,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2048537998ea5d7611dce0181d5443c2b4398a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.866ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(S,X)}"></span>가 <a href="/wiki/%EC%9C%A0%EA%B3%84_%ED%95%A8%EC%88%98" title="유계 함수">유계 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5566629251250ba644683a256f3ae6b6ec516d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle S\to X}"></span>들의 집합이라고 하자. 이 위에 다음과 같은 <a href="/wiki/%EC%83%81%ED%95%9C" class="mw-redirect" title="상한">상한</a> <a href="/wiki/%EA%B1%B0%EB%A6%AC_%ED%95%A8%EC%88%98" title="거리 함수">거리 함수</a>를 주자. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f,g)=\sup _{s\in S}d\left(f(s),g(s)\right)\qquad (f,g\in {\mathcal {B}}(S,X))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g)=\sup _{s\in S}d\left(f(s),g(s)\right)\qquad (f,g\in {\mathcal {B}}(S,X))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41bea7a36e0988f595c824264e93486bd2a3dec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.057ex; height:4.509ex;" alt="{\displaystyle d(f,g)=\sup _{s\in S}d\left(f(s),g(s)\right)\qquad (f,g\in {\mathcal {B}}(S,X))}"></span></dd></dl> <p>그렇다면 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(S,X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(S,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2048537998ea5d7611dce0181d5443c2b4398a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.866ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(S,X)}"></span>는 완비 거리 공간을 이룬다. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26858958"><div class="proof mw-collapsible mw-collapsed"> <div class="prooftitle"> <p><span class="prooftitletext">증명:</span> </p> </div> <div class="proofcontent mw-collapsible-content"> <p>임의의 코시 점렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i})_{i=0}^{\infty }\subseteq {\mathcal {B}}(S,X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i=0}^{\infty }\subseteq {\mathcal {B}}(S,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc647ad73e59a3060d3a7c5b09442a0e9c7ef982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.613ex; height:3.009ex;" alt="{\displaystyle (f_{i})_{i=0}^{\infty }\subseteq {\mathcal {B}}(S,X)}"></span>이 수렴함을 보이는 것으로 충분하다. 각 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acce52dffd84d073a24f4606a175da60148fd0c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i}(s))_{i=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i}(s))_{i=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2453d8309a2ca15b47becf45660d06249874d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.548ex; height:3.009ex;" alt="{\displaystyle (f_{i}(s))_{i=0}^{\infty }}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 위의 코시 점렬이므로 어떤 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(s)\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(s)\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277347e1e6ecdd9e552ee24b98fb541676f384f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.999ex; height:2.843ex;" alt="{\displaystyle f(s)\in X}"></span>로 수렴하며, 이 경우 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon S\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon S\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0616a9592161e099006364af8a65b7d8d5a5d15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.406ex; height:2.509ex;" alt="{\displaystyle f\colon S\to X}"></span>는 <a href="/wiki/%EC%9C%A0%EA%B3%84_%ED%95%A8%EC%88%98" title="유계 함수">유계 함수</a>이다. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i})_{i=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7350f2292537377914c755bdda092c8341599617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.648ex; height:3.009ex;" alt="{\displaystyle (f_{i})_{i=0}^{\infty }}"></span>가 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>로 수렴함은 다음과 같이 보일 수 있다. 임의의 양의 실수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i})_{i=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7350f2292537377914c755bdda092c8341599617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.648ex; height:3.009ex;" alt="{\displaystyle (f_{i})_{i=0}^{\infty }}"></span>가 코시 점렬이므로 다음 조건을 만족시키는 <a href="/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수">자연수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\epsilon )\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\epsilon )\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616334231c5269acffa23d9df6d3f213884f7ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.336ex; height:2.843ex;" alt="{\displaystyle N(\epsilon )\in \mathbb {N} }"></span>가 존재한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f_{i}(s),f_{j}(s))<\epsilon \qquad \forall s\in S,\;i,j\geq N(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo><</mo> <mi>ϵ</mi> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≥</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f_{i}(s),f_{j}(s))<\epsilon \qquad \forall s\in S,\;i,j\geq N(\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427d7a5383a86880ec2ad667764cecb0f7999ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.646ex; height:3.009ex;" alt="{\displaystyle d(f_{i}(s),f_{j}(s))<\epsilon \qquad \forall s\in S,\;i,j\geq N(\epsilon )}"></span></dd></dl> <p>위 조건에서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e8cf365e05144f87bbe5a66b2a3ccbaa1a8191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:6.923ex; height:2.509ex;" alt="{\displaystyle j\to \infty }"></span>를 취하면 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f_{i}(s),f(s))\leq \epsilon \qquad \forall s\in S,\;i\geq N(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ϵ</mi> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>≥</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f_{i}(s),f(s))\leq \epsilon \qquad \forall s\in S,\;i\geq N(\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ecca4303b9004b92be579d833dfa01171dd9b22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.884ex; height:2.843ex;" alt="{\displaystyle d(f_{i}(s),f(s))\leq \epsilon \qquad \forall s\in S,\;i\geq N(\epsilon )}"></span></dd></dl> <p>을 얻는다. 즉, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f_{i},f)\leq \epsilon \qquad \forall i\geq N(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ϵ</mi> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>≥</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f_{i},f)\leq \epsilon \qquad \forall i\geq N(\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed212ed489ced4e88eacfc220b16d6dbf9875eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.975ex; height:2.843ex;" alt="{\displaystyle d(f_{i},f)\leq \epsilon \qquad \forall i\geq N(\epsilon )}"></span></dd></dl> <p>이다. </p> </div></div> <p><a href="/wiki/%EC%9C%84%EC%83%81_%EA%B3%B5%EA%B0%84_(%EC%88%98%ED%95%99)" title="위상 공간 (수학)">위상 공간</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> 및 완비 거리 공간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {CB}}(S,X)\subset {\mathcal {B}}(S,X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {CB}}(S,X)\subset {\mathcal {B}}(S,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b04b769e56dcd94c5dda1c35182c5c076fa467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.069ex; height:2.843ex;" alt="{\displaystyle {\mathcal {CB}}(S,X)\subset {\mathcal {B}}(S,X)}"></span>가 <a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속</a> <a href="/wiki/%EC%9C%A0%EA%B3%84_%ED%95%A8%EC%88%98" title="유계 함수">유계 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5566629251250ba644683a256f3ae6b6ec516d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle S\to X}"></span>들의 집합이라고 하자. 이는 상한 거리 함수에 대하여 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>을 이루며, 따라서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {CB}}(S,X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {CB}}(S,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd30a2ccb9aa1e38e43a10b3beaa4658ac073d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.105ex; height:2.843ex;" alt="{\displaystyle {\mathcal {CB}}(S,X)}"></span> 역시 완비 거리 공간을 이룬다. </p> <div class="mw-heading mw-heading3"><h3 id="바나흐_공간"><span id=".EB.B0.94.EB.82.98.ED.9D.90_.EA.B3.B5.EA.B0.84"></span>바나흐 공간</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=15" title="부분 편집: 바나흐 공간"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%B0%94%EB%82%98%ED%9D%90_%EA%B3%B5%EA%B0%84" title="바나흐 공간">바나흐 공간</a>입니다.</div> <p><a href="/wiki/%EB%85%B8%EB%A6%84_%EA%B3%B5%EA%B0%84" title="노름 공간">노름 공간</a> 가운데 완비 거리 공간을 이루는 것을 <b><a href="/wiki/%EB%B0%94%EB%82%98%ED%9D%90_%EA%B3%B5%EA%B0%84" title="바나흐 공간">바나흐 공간</a></b>이라고 한다. 마찬가지로, <a href="/wiki/%EB%82%B4%EC%A0%81_%EA%B3%B5%EA%B0%84" title="내적 공간">내적 공간</a> 가운데 완비 거리 공간을 이루는 것을 <b><a href="/wiki/%ED%9E%90%EB%B2%A0%EB%A5%B4%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="힐베르트 공간">힐베르트 공간</a></b>이라고 한다. </p> <div class="mw-heading mw-heading2"><h2 id="역사"><span id=".EC.97.AD.EC.82.AC"></span>역사</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=16" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>역사적으로, 코시 점렬의 개념은 <a href="/wiki/%EC%88%98%EC%97%B4" title="수열">수열</a>의 <a href="/wiki/%EA%B7%B9%ED%95%9C" title="극한">극한</a>과 <a href="/wiki/%EA%B8%89%EC%88%98_(%EC%88%98%ED%95%99)" title="급수 (수학)">급수</a>의 개념을 엄밀하게 정의하려는 시도에서 비롯되었다. 1817년에 <a href="/wiki/%EB%B2%A0%EB%A5%B4%EB%82%98%EB%A5%B4%ED%8A%B8_%EB%B3%BC%EC%B0%A8%EB%85%B8" title="베르나르트 볼차노">베르나르트 볼차노</a>는 <a href="/wiki/%EC%A4%91%EA%B0%84%EA%B0%92_%EC%A0%95%EB%A6%AC" title="중간값 정리">중간값 정리</a>에 대한 논문<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>에서 코시 점렬의 개념을 사용하였으나,<sup id="cite_ref-Lutzen_3-0" class="reference"><a href="#cite_note-Lutzen-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:§6.4.2, 174–176</sup></span> 서유럽에서 멀리 떨어진 <a href="/wiki/%ED%94%84%EB%9D%BC%ED%95%98" title="프라하">프라하</a>에서 살던 볼차노의 업적은 당시 널리 주목받지 못했다. 이후 <a href="/wiki/%EC%98%A4%EA%B7%80%EC%8A%A4%ED%83%B1_%EB%A3%A8%EC%9D%B4_%EC%BD%94%EC%8B%9C" title="오귀스탱 루이 코시">오귀스탱 루이 코시</a>가 1921년에 유명한 저서 《<a href="/wiki/%EC%97%90%EC%BD%9C_%ED%8F%B4%EB%A6%AC%ED%85%8C%ED%81%AC%EB%8B%88%ED%81%AC" title="에콜 폴리테크니크">에콜 폴리테크니크</a> 해석학 교재》(<span style="font-size: smaller;"><a href="/wiki/%ED%94%84%EB%9E%91%EC%8A%A4%EC%96%B4" title="프랑스어">프랑스어</a>: </span><span lang="fr">Cours d'Analyse de l’École Royale Polytechnique</span>)<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>에서 <a href="/wiki/%EA%B8%89%EC%88%98_(%EC%88%98%ED%95%99)" title="급수 (수학)">급수</a>의 수렴에 대한 조건을 정의하기 위하여 같은 개념을 사용하였다.<sup id="cite_ref-Lutzen_3-1" class="reference"><a href="#cite_note-Lutzen-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:§6.3.4, 167</sup></span> </p><p>카리스티 고정점 정리는 제임스 카리스티(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">James V. Caristi</span>)가 1976년 논문에서 제시하였다.<sup id="cite_ref-Caristi_5-0" class="reference"><a href="#cite_note-Caristi-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> 카리스티는 증명에서 <a href="/wiki/%EC%B4%88%ED%95%9C_%EA%B7%80%EB%82%A9%EB%B2%95" class="mw-redirect" title="초한 귀납법">초한 귀납법</a>을 사용하였으며, 이는 <a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a>에 의존한다. 이후 더 약한 조건인 <a href="/wiki/%EC%9D%98%EC%A1%B4%EC%A0%81_%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" class="mw-redirect" title="의존적 선택 공리">의존적 선택 공리</a>에 의존하는 방법들로 재증명되었다. 로만 만카(<span style="font-size: smaller;"><a href="/wiki/%ED%8F%B4%EB%9E%80%EB%93%9C%EC%96%B4" title="폴란드어">폴란드어</a>: </span><span lang="pl">Roman Manka</span>)가 1988년 논문에서 어떠한 꼴의 선택 공리도 필요 없는 증명을 제시하였다.<sup id="cite_ref-Mańka_6-0" class="reference"><a href="#cite_note-Mańka-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> 카리스티 고정점 정리를 완비 거리 공간의 정의로 삼을 수 있다는 사실은 카리스티의 지도 교수였던 윌리엄 아서 커크(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">William Arthur Kirk</span>)가 1976년 논문에서 증명하였다.<sup id="cite_ref-Kirk_7-0" class="reference"><a href="#cite_note-Kirk-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>칸난 고정점 정리는 라빈드란 칸난(<span style="font-size: smaller;"><a href="/wiki/%ED%83%80%EB%B0%80%EC%96%B4" title="타밀어">타밀어</a>: </span><span lang="ta">ரவிந்திரன் கண்ணன்</span>, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Ravindran Kannan</span>)이 1968년 논문에서 증명하였다.<sup id="cite_ref-Kannan_8-0" class="reference"><a href="#cite_note-Kannan-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> 수브라마냠(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">P. V. Subrahmanyam</span>)<sup id="cite_ref-Subrahmanyam_9-0" class="reference"><a href="#cite_note-Subrahmanyam-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>과 시오지 나오키(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Naoki Shioji</span>), 스즈키 도모나리(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Tomonari Suzuki</span>), 다카하시 와타루(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Wataru Takahashi</span>)<sup id="cite_ref-Shioji_10-0" class="reference"><a href="#cite_note-Shioji-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>가 칸난 고정점 정리를 완비 거리 공간의 정의로 삼을 수 있음을 독자적으로 증명하였다. </p> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=17" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Grunbaum-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Grunbaum_1-0">가</a></sup> <sup><a href="#cite_ref-Grunbaum_1-1">나</a></sup> <sup><a href="#cite_ref-Grunbaum_1-2">다</a></sup></span> <span class="reference-text"><cite class="citation journal">Grünbaum, B. (1960). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.pjm/1103038634">“Some applications of expansion constants”</a>. 《Pacific Journal of Mathematics》 (영어) <b>10</b> (1): 193–201. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0114162">0114162</a>. <a href="/wiki/%EC%B2%B8%ED%8A%B8%EB%9E%84%EB%B8%94%EB%9D%BC%ED%8A%B8_%EB%A7%88%ED%8A%B8" title="첸트랄블라트 마트">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0094.09002">0094.09002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pacific+Journal+of+Mathematics&rft.atitle=Some+applications+of+expansion+constants&rft.volume=10&rft.issue=1&rft.pages=193-201&rft.date=1960&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0094.09002&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0114162&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=B.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.pjm%2F1103038634&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/%EB%B2%A0%EB%A5%B4%EB%82%98%EB%A5%B4%ED%8A%B8_%EB%B3%BC%EC%B0%A8%EB%85%B8" title="베르나르트 볼차노">Bolzano, Bernard</a> (1817). “Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege” (독일어). Wilhelm Engelmann.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Rein+analytischer+Beweis+des+Lehrsatzes%2C+dass+zwischen+je+zwey+Werthen%2C+die+ein+entgegengesetzes+Resultat+gew%C3%A4hren%2C+wenigstens+eine+reele+Wurzel+der+Gleichung+liege&rft.date=1817&rft.aulast=Bolzano&rft.aufirst=Bernard&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Lutzen-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Lutzen_3-0">가</a></sup> <sup><a href="#cite_ref-Lutzen_3-1">나</a></sup></span> <span class="reference-text"><cite class="citation book">Lützen, Jesper (2003). 〈The foundation of analysis in the 19th century〉. Hans Niels Jahnke. <a rel="nofollow" class="external text" href="http://www.ams.org/bookstore-getitem/item=HMATH-24">《A history of analysis》</a>. History of Mathematics (영어) <b>24</b>. American Mathematical Society, London Mathematical Society. 155–212쪽. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-0-8218-2623-2" title="특수:책찾기/978-0-8218-2623-2"><bdi>978-0-8218-2623-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+foundation+of+analysis+in+the+19th+century&rft.btitle=A+history+of+analysis&rft.series=History+of+Mathematics&rft.pages=155-212&rft.pub=American+Mathematical+Society%2C+London+Mathematical+Society&rft.date=2003&rft.isbn=978-0-8218-2623-2&rft.aulast=L%C3%BCtzen&rft.aufirst=Jesper&rft_id=http%3A%2F%2Fwww.ams.org%2Fbookstore-getitem%2Fitem%3DHMATH-24&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/%EC%98%A4%EA%B7%80%EC%8A%A4%ED%83%B1_%EB%A3%A8%EC%9D%B4_%EC%BD%94%EC%8B%9C" title="오귀스탱 루이 코시">Cauchy, Augustin-Louis</a> (1821). 《Cours d’Analyse de l’Ecole royale polytechnique. 1. Analyse Algébrique》 (프랑스어). L’Imprimerie Royale, Debure frères, Libraires du Roi et de la Bibliothèque du Roi.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cours+d%E2%80%99Analyse+de+l%E2%80%99Ecole+royale+polytechnique.+1.+Analyse+Alg%C3%A9brique&rft.pub=L%E2%80%99Imprimerie+Royale%2C+Debure+fr%C3%A8res%2C+Libraires+du+Roi+et+de+la+Biblioth%C3%A8que+du+Roi&rft.date=1821&rft.aulast=Cauchy&rft.aufirst=Augustin-Louis&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Caristi-5"><span class="mw-cite-backlink"><a href="#cite_ref-Caristi_5-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Caristi, James (1976). “Fixed point theorems for mappings satisfying inwardness conditions”. 《Transactions of the American Mathematical Society》 (영어) <b>215</b>: 241–251. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0002-9947-1976-0394329-4">10.1090/S0002-9947-1976-0394329-4</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0002-9947">0002-9947</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0394329">0394329</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Fixed+point+theorems+for+mappings+satisfying+inwardness+conditions&rft.volume=215&rft.pages=241-251&rft.date=1976&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0394329&rft.issn=0002-9947&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1976-0394329-4&rft.aulast=Caristi&rft.aufirst=James&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Mańka-6"><span class="mw-cite-backlink"><a href="#cite_ref-Mańka_6-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Mańka, Roman (1988). “Some forms of the axiom of choice”. 《Jahrbuch der Kurt-Gödel-Gesellschaft》 (영어) <b>1</b>: 24–34.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Jahrbuch+der+Kurt-G%C3%B6del-Gesellschaft&rft.atitle=Some+forms+of+the+axiom+of+choice&rft.volume=1&rft.pages=24-34&rft.date=1988&rft.aulast=Ma%C5%84ka&rft.aufirst=Roman&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Kirk-7"><span class="mw-cite-backlink"><a href="#cite_ref-Kirk_7-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Kirk, William Arthur (1976). “Caristi’s fixed point theorem and metric convexity”. 《Colloquium Mathematicum》 (영어) <b>36</b>: 81–86. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1186%2Fs13663-015-0464-5">10.1186/s13663-015-0464-5</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0010-1354">0010-1354</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Colloquium+Mathematicum&rft.atitle=Caristi%E2%80%99s+fixed+point+theorem+and+metric+convexity&rft.volume=36&rft.pages=81-86&rft.date=1976&rft_id=info%3Adoi%2F10.1186%2Fs13663-015-0464-5&rft.issn=0010-1354&rft.aulast=Kirk&rft.aufirst=William+Arthur&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Kannan-8"><span class="mw-cite-backlink"><a href="#cite_ref-Kannan_8-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Kannan, Ravindran (1968). “Some results on fixed points”. 《Bulletin of the Calcutta Mathematical Society》 (영어) <b>60</b>: 71–76. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0008-0659">0008-0659</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=257837">257837</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+Calcutta+Mathematical+Society&rft.atitle=Some+results+on+fixed+points&rft.volume=60&rft.pages=71-76&rft.date=1968&rft.issn=0008-0659&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D257837&rft.aulast=Kannan&rft.aufirst=Ravindran&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Subrahmanyam-9"><span class="mw-cite-backlink"><a href="#cite_ref-Subrahmanyam_9-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Subrahmanyam, P. V. (1975). “Completeness and fixed-points”. 《Monatshefte für Mathematik》 (영어) <b>80</b>: 325–330. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF01472580">10.1007/BF01472580</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0026-9255">0026-9255</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatshefte+f%C3%BCr+Mathematik&rft.atitle=Completeness+and+fixed-points&rft.volume=80&rft.pages=325-330&rft.date=1975&rft_id=info%3Adoi%2F10.1007%2FBF01472580&rft.issn=0026-9255&rft.aulast=Subrahmanyam&rft.aufirst=P.+V.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Shioji-10"><span class="mw-cite-backlink"><a href="#cite_ref-Shioji_10-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Shioji, Naoki; Suzuki, Tomonari; Takahashi, Wataru (1998). “Contractive mappings, Kannan mappings and metric completeness”. 《Proceedings of the American Mathematical Society》 (영어) <b>126</b>: 3117–3124. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0002-9939-98-04605-X">10.1090/S0002-9939-98-04605-X</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0002-9939">0002-9939</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1469434">1469434</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Contractive+mappings%2C+Kannan+mappings+and+metric+completeness&rft.volume=126&rft.pages=3117-3124&rft.date=1998&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1469434&rft.issn=0002-9939&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-98-04605-X&rft.aulast=Shioji&rft.aufirst=Naoki&rft.au=Suzuki%2C+Tomonari&rft.au=Takahashi%2C+Wataru&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%99%84%EB%B9%84_%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84&action=edit&section=18" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Complete_metric_space">“Complete metric space”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Complete+metric+space&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FComplete_metric_space&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Cauchy_sequence">“Cauchy sequence”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Cauchy+sequence&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FCauchy_sequence&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Cauchy_criteria">“Cauchy criteria”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Cauchy+criteria&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FCauchy_criteria&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CompleteMetricSpace.html">“Complete metric space”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Complete+metric+space&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCompleteMetricSpace.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CauchySequence.html">“Cauchy sequence”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Cauchy+sequence&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCauchySequence.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/complete+space">“Complete space”</a>. 《nLab》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Complete+space&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fcomplete%2Bspace&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/complete+topological+space">“Complete topological space”</a>. 《nLab》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Complete+topological+space&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fcomplete%2Btopological%2Bspace&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20151210221206/https://proofwiki.org/wiki/Definition:Cauchy_Sequence">“Definition: Cauchy sequence”</a>. 《ProofWiki》 (영어). 2015년 12월 10일에 <a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Cauchy_Sequence">원본 문서</a>에서 보존된 문서<span class="reference-accessdate">. 2015년 12월 10일에 확인함</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ProofWiki&rft.atitle=Definition%3A+Cauchy+sequence&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FDefinition%3ACauchy_Sequence&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%99%84%EB%B9%84+%EA%B1%B0%EB%A6%AC+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20151210220627/https://proofwiki.org/wiki/Definition:Complete_Metric_Space">“Definition: complete metric space”</a>. 《ProofWiki》 (영어). 2015년 12월 10일에 <a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Complete_Metric_Space">원본 문서</a>에서 보존된 문서<span class="reference-accessdate">. 2015년 12월 10일에 확인함</span>.</cite><span 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