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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%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" 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%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" 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%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" 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-유클리드_기하학" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#유클리드_기하학"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>유클리드 기하학</span> </div> </a> <button aria-controls="toc-유클리드_기하학-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>유클리드 기하학 하위섹션 토글하기</span> </button> <ul id="toc-유클리드_기하학-sublist" class="vector-toc-list"> <li id="toc-절선_부등식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#절선_부등식"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>절선 부등식</span> </div> </a> <ul id="toc-절선_부등식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-고차원_단체_부등식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#고차원_단체_부등식"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>고차원 단체 부등식</span> </div> </a> <ul id="toc-고차원_단체_부등식-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-노름_벡터_공간" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#노름_벡터_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>노름 벡터 공간</span> </div> </a> <ul id="toc-노름_벡터_공간-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-거리_공간" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#거리_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>거리 공간</span> </div> </a> <ul id="toc-거리_공간-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-역삼각_부등식" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역삼각_부등식"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>역삼각 부등식</span> </div> </a> <ul id="toc-역삼각_부등식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-민코프스키_공간에서의_부등호_반전" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#민코프스키_공간에서의_부등호_반전"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>민코프스키 공간에서의 부등호 반전</span> </div> </a> <ul id="toc-민코프스키_공간에서의_부등호_반전-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">6</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-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">7</span> <span>각주</span> </div> </a> <button aria-controls="toc-각주-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>각주 하위섹션 토글하기</span> </button> <ul id="toc-각주-sublist" class="vector-toc-list"> <li id="toc-내용주" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#내용주"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>내용주</span> </div> </a> <ul id="toc-내용주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-참고주" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#참고주"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>참고주</span> </div> </a> <ul id="toc-참고주-sublist" class="vector-toc-list"> </ul> </li> </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">8</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-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">삼각 부등식</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. 44개 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-44" 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">44개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Dreiecksungleichung" title="Dreiecksungleichung – 독일어(스위스)" lang="gsw" hreflang="gsw" data-title="Dreiecksungleichung" data-language-autonym="Alemannisch" data-language-local-name="독일어(스위스)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%A8%D8%A7%D9%8A%D9%86%D8%A9_%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB" title="متباينة المثلث – 아랍어" lang="ar" hreflang="ar" data-title="متباينة المثلث" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D1%8F%D1%80%D0%BE%D1%9E%D0%BD%D0%B0%D1%81%D1%86%D1%8C_%D1%82%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA%D0%B0" title="Няроўнасць трохвугольніка – 벨라루스어" lang="be" hreflang="be" data-title="Няроўнасць трохвугольніка" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%BD%D0%B0_%D1%82%D1%80%D0%B8%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%B8%D0%BA%D0%B0" title="Неравенство на триъгълника – 불가리아어" lang="bg" hreflang="bg" data-title="Неравенство на триъгълника" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Desigualtat_triangular" title="Desigualtat triangular – 카탈로니아어" lang="ca" hreflang="ca" data-title="Desigualtat triangular" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%D8%A7%D8%B3%DB%95%D9%86%DA%AF%DB%95%DB%8C_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C" title="لاسەنگەی سێگۆشەیی – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="لاسەنگەی سێگۆشەیی" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Troj%C3%BAheln%C3%ADkov%C3%A1_nerovnost" title="Trojúhelníková nerovnost – 체코어" lang="cs" hreflang="cs" data-title="Trojúhelníková nerovnost" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B8%C3%A7%D0%BA%C4%95%D1%82%D0%B5%D1%81%D0%BB%C4%95%D1%85_%D1%82%D0%B0%D0%BD%D0%BC%D0%B0%D1%80%D0%BB%C4%83%D1%85%C4%95" title="Виçкĕтеслĕх танмарлăхĕ – 추바시어" lang="cv" hreflang="cv" data-title="Виçкĕтеслĕх танмарлăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trekantsuligheden" title="Trekantsuligheden – 덴마크어" lang="da" hreflang="da" data-title="Trekantsuligheden" 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/Dreiecksungleichung" title="Dreiecksungleichung – 독일어" lang="de" hreflang="de" data-title="Dreiecksungleichung" 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%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%B9%CF%83%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Τριγωνική ανισότητα – 그리스어" lang="el" hreflang="el" data-title="Τριγωνική ανισότητα" 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/Triangle_inequality" title="Triangle inequality – 영어" lang="en" hreflang="en" data-title="Triangle inequality" 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/Neegala%C4%B5o_de_triangulo" title="Neegalaĵo de triangulo – 에스페란토어" lang="eo" hreflang="eo" data-title="Neegalaĵo de triangulo" 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/Desigualdad_triangular" title="Desigualdad triangular – 스페인어" lang="es" hreflang="es" data-title="Desigualdad triangular" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kolmnurga_v%C3%B5rratus" title="Kolmnurga võrratus – 에스토니아어" lang="et" hreflang="et" data-title="Kolmnurga võrratus" data-language-autonym="Eesti" data-language-local-name="에스토니아어" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Desberdintza_triangeluar" title="Desberdintza triangeluar – 바스크어" lang="eu" hreflang="eu" data-title="Desberdintza triangeluar" data-language-autonym="Euskara" data-language-local-name="바스크어" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%A7%D8%A8%D8%B1%D8%A7%D8%A8%D8%B1%DB%8C_%D9%85%D8%AB%D9%84%D8%AB%DB%8C" title="نابرابری مثلثی – 페르시아어" lang="fa" hreflang="fa" data-title="نابرابری مثلثی" 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/Kolmioep%C3%A4yht%C3%A4l%C3%B6" title="Kolmioepäyhtälö – 핀란드어" lang="fi" hreflang="fi" data-title="Kolmioepäyhtälö" 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/In%C3%A9galit%C3%A9_triangulaire" title="Inégalité triangulaire – 프랑스어" lang="fr" hreflang="fr" data-title="Inégalité triangulaire" 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/Desigualdade_triangular" title="Desigualdade triangular – 갈리시아어" lang="gl" hreflang="gl" data-title="Desigualdade triangular" 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%90%D7%99-%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%94%D7%9E%D7%A9%D7%95%D7%9C%D7%A9" title="אי-שוויון המשולש – 히브리어" lang="he" hreflang="he" data-title="אי-שוויון המשולש" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C_%E0%A4%85%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A4%BE" title="त्रिभुज असमिका – 힌디어" lang="hi" hreflang="hi" data-title="त्रिभुज असमिका" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C3%A1romsz%C3%B6g-egyenl%C5%91tlens%C3%A9g" title="Háromszög-egyenlőtlenség – 헝가리어" lang="hu" hreflang="hu" data-title="Háromszög-egyenlőtlenség" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%A1%D5%B6_%D5%A1%D5%B6%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Եռանկյան անհավասարություն – 아르메니아어" lang="hy" hreflang="hy" data-title="Եռանկյան անհավասարություն" 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/Pertidaksamaan_segitiga" title="Pertidaksamaan segitiga – 인도네시아어" lang="id" hreflang="id" data-title="Pertidaksamaan segitiga" 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/%C3%9Er%C3%ADhyrnings%C3%B3jafna" title="Þríhyrningsójafna – 아이슬란드어" lang="is" hreflang="is" data-title="Þríhyrningsójafna" 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/Disuguaglianza_triangolare" title="Disuguaglianza triangolare – 이탈리아어" lang="it" hreflang="it" data-title="Disuguaglianza triangolare" 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/%E4%B8%89%E8%A7%92%E4%B8%8D%E7%AD%89%E5%BC%8F" title="三角不等式 – 일본어" lang="ja" hreflang="ja" data-title="三角不等式" 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-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Disegualiansa_triangol%C3%A2" title="Disegualiansa triangolâ – Ligurian" lang="lij" hreflang="lij" data-title="Disegualiansa triangolâ" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trikampio_nelygyb%C4%97" title="Trikampio nelygybė – 리투아니아어" lang="lt" hreflang="lt" data-title="Trikampio nelygybė" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Driehoeksongelijkheid" title="Driehoeksongelijkheid – 네덜란드어" lang="nl" hreflang="nl" data-title="Driehoeksongelijkheid" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trekantulikskapen" title="Trekantulikskapen – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Trekantulikskapen" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Nier%C3%B3wno%C5%9B%C4%87_tr%C3%B3jk%C4%85ta" title="Nierówność trójkąta – 폴란드어" lang="pl" hreflang="pl" data-title="Nierówność trójkąta" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Desigualdade_triangular" title="Desigualdade triangular – 포르투갈어" lang="pt" hreflang="pt" data-title="Desigualdade triangular" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Inegalitatea_triunghiului" title="Inegalitatea triunghiului – 루마니아어" lang="ro" hreflang="ro" data-title="Inegalitatea triunghiului" data-language-autonym="Română" data-language-local-name="루마니아어" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%D0%B0" title="Неравенство треугольника – 러시아어" lang="ru" hreflang="ru" data-title="Неравенство треугольника" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B5%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D0%BA%D0%BE%D1%81%D1%82_%D1%82%D1%80%D0%BE%D1%83%D0%B3%D0%BB%D0%B0" title="Неједнакост троугла – 세르비아어" lang="sr" hreflang="sr" data-title="Неједнакост троугла" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Triangelolikheten" title="Triangelolikheten – 스웨덴어" lang="sv" hreflang="sv" data-title="Triangelolikheten" data-language-autonym="Svenska" data-language-local-name="스웨덴어" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%9F%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9F%E0%AF%88_%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88" title="அடிப்படை முக்கோணச் சமனின்மை – 타밀어" lang="ta" hreflang="ta" data-title="அடிப்படை முக்கோணச் சமனின்மை" data-language-autonym="தமிழ்" data-language-local-name="타밀어" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9C%C3%A7gen_e%C5%9Fitsizli%C4%9Fi" title="Üçgen eşitsizliği – 터키어" lang="tr" hreflang="tr" data-title="Üçgen eşitsizliği" data-language-autonym="Türkçe" data-language-local-name="터키어" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D1%96%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D1%82%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA%D0%B0" title="Нерівність трикутника – 우크라이나어" lang="uk" hreflang="uk" data-title="Нерівність трикутника" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/B%E1%BA%A5t_%C4%91%E1%BA%B3ng_th%E1%BB%A9c_tam_gi%C3%A1c" title="Bất đẳng thức tam giác – 베트남어" lang="vi" hreflang="vi" data-title="Bất đẳng thức tam giác" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E4%B8%8D%E7%AD%89%E5%BC%8F" title="三角不等式 – 중국어" lang="zh" hreflang="zh" data-title="三角不等式" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E4%B8%8D%E7%AD%89%E5%BC%8F" title="三角不等式 – 광둥어" lang="yue" hreflang="yue" data-title="三角不等式" data-language-autonym="粵語" data-language-local-name="광둥어" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q208216#sitelinks-wikipedia" title="언어 간 링크 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href="https://commons.wikimedia.org/wiki/Category:Triangle_inequality" hreflang="en"><span>위키미디어 공용</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q208216" title="데이터 저장소에 연결된 항목을 가리키는 링크 [g]" accesskey="g"><span>위키데이터 항목</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="페이지 도구"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="보이기"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:TriangleInequality.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/TriangleInequality.svg/250px-TriangleInequality.svg.png" decoding="async" width="250" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/TriangleInequality.svg/375px-TriangleInequality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/TriangleInequality.svg/500px-TriangleInequality.svg.png 2x" data-file-width="938" data-file-height="906" /></a><figcaption>세 변의 길이를 <style data-mw-deduplicate="TemplateStyles:r25030363">.mw-parser-output .texhtml{font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;white-space:nowrap;line-height:1;font-size:118%}</style><span class="texhtml" style="font-style: italic;">x, y, z</span>로 하는 삼각형의 3가지 예시</figcaption></figure> <p><b>삼각 부등식</b>(三角不等式, Triangle inequality)은 <a href="/wiki/%EC%88%98%ED%95%99" title="수학">수학</a>에서 <a href="/wiki/%EC%82%BC%EA%B0%81%ED%98%95" title="삼각형">삼각형</a>의 세 변에 대한 <a href="/wiki/%EB%B6%80%EB%93%B1%EC%8B%9D" title="부등식">부등식</a>이다. 이 부등식은 임의의 삼각형에 대하여 그 임의의 두 변의 합이 나머지 한 변보다 커야 함을 말하는 것으로서 <a href="/wiki/%EA%B8%B0%ED%95%98%ED%95%99" title="기하학">기하학</a>의 여러 공간에 적용된다.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> </p><p>삼각형의 세 변이 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x, y, z</span>에서 최대 변이 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">z</span>라고 하면 삼각 부등식은 <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 z\leq x+y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>≤</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\leq x+y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdb8951da065694a53a81654ce73ecff8cfbd9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.512ex; height:2.343ex;" alt="{\displaystyle z\leq x+y}"></span>이 성립됨을 주장하고 있다.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>주해 1<span class="cite-bracket">]</span></a></sup> <a href="/wiki/%EB%93%B1%ED%98%B8" title="등호">등호</a>가 성립하는 것은 삼각형이 면적 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">0</span>으로 퇴화한 때에 한한다. <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B8%B0%ED%95%98%ED%95%99" title="유클리드 기하학">유클리드 기하학</a> 이외의 몇 개의 기하학에서 삼각 부등식은 거리에 관한 정리로서 벡터나 벡터의 길이(노름)를 이용하여 <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 \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74326462abceef6a4197bcf12093abe3bb90524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.398ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|}"></span>라고 표현할 수 있다. 여기서 3번째 변 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">z</span>의 길이가 벡터의 합 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><b>x</b> + <b>y</b></span>로 치환되고 있는 것에 주의해야 한다. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x, y</span>가 <a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a>일 때에 그것을 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">ℝ<sup>1</sup></span>의 벡터로 본다면 삼각 부등식은 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92" title="절댓값">절댓값</a> 사이의 관계를 서술하는 것이 된다. </p><p>유클리드 기하학에서 <a href="/wiki/%EC%A7%81%EA%B0%81%EC%82%BC%EA%B0%81%ED%98%95" title="직각삼각형">직각삼각형</a>에 대한 삼각 부등식은 <a href="/wiki/%ED%94%BC%ED%83%80%EA%B3%A0%EB%9D%BC%EC%8A%A4%EC%9D%98_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="피타고라스의 정리">피타고라스의 정리</a>의 귀결이며 일반 삼각형의 경우에는 <a href="/wiki/%EC%BD%94%EC%82%AC%EC%9D%B8_%EB%B2%95%EC%B9%99" title="코사인 법칙">코사인 법칙</a>의 귀결인데 그러한 정리에 의하지 않는 증명은 가능하다. 삼각 부등식은 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">ℝ<sup>2</sup></span>이나 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">ℝ<sup>3</sup></span> 가운데 어느 곳에서 직관적으로 볼 수 있다. 그림은 분명히 부등호가 성립되는 것(위쪽)부터 등호에 가까운 것(아래쪽)까지의 3가지 예시이다. 유클리드 기하학의 경우 등호가 성립하려면 하나의 각이 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">180°</span>이고 2개의 각이 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">0°</span>인 경우, 따라서 3개의 꼭짓점이 동일 직선상에 있는 경우(<a href="/wiki/%EA%B3%B5%EC%84%A0%EC%A0%90" title="공선점">공선점</a>)에 한정된다. 따라서 유클리드 기하학에서 2개의 점 사이의 최단 거리는 직선이다. </p><p><a href="/wiki/%EA%B5%AC%EB%A9%B4%EA%B8%B0%ED%95%98%ED%95%99" title="구면기하학">구면기하학</a>에서 2개의 점 사이의 최단 거리는 <a href="/wiki/%EB%8C%80%EC%9B%90" title="대원">대원</a>의 호이지만 구면상의 2개의 점 사이의 거리가 그 2개의 점을 연결하는 열호선분(대원 안에서 그 2개의 점을 끝점으로 하는 2개의 호 가운데 중심각이 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">[0, <i>π</i>)</span>인 것)으로 주어지는 것이라고 한다면 삼각 부등식이 성립된다.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>삼각 부등식은 노름이나 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%ED%95%A8%EC%88%98" title="거리 함수">거리 함수</a>의 '정의 성질' 가운데 하나이다. 그러한 성질은 각각 특정 공간(<a href="/wiki/%EC%8B%A4%EC%A7%81%EC%84%A0" title="실직선">실직선</a>이나 <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간">유클리드 공간</a>이나 (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>p</i> ≥ 1</span>에 대한) <a href="/wiki/%EB%A5%B4%EB%B2%A0%EA%B7%B8_%EA%B3%B5%EA%B0%84" title="르베그 공간">르베그 공간</a>(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">L<sup>p</sup></span>-공간)이나 <a href="/wiki/%EB%82%B4%EC%A0%81_%EA%B3%B5%EA%B0%84" title="내적 공간">내적 공간</a>)에 대해 그러한 노름이나 거리 함수가 되어야 하는 임의의 함수에 대한 정리로서 제대로 서술하지 않으면 안 된다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="유클리드_기하학"><span id=".EC.9C.A0.ED.81.B4.EB.A6.AC.EB.93.9C_.EA.B8.B0.ED.95.98.ED.95.99"></span>유클리드 기하학</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=1" title="부분 편집: 유클리드 기하학"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Euclid_triangle_inequality.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Euclid_triangle_inequality.svg/250px-Euclid_triangle_inequality.svg.png" decoding="async" width="250" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Euclid_triangle_inequality.svg/375px-Euclid_triangle_inequality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Euclid_triangle_inequality.svg/500px-Euclid_triangle_inequality.svg.png 2x" data-file-width="535" data-file-height="417" /></a><figcaption>유클리드 평면 기하의 삼각 부등식 증명의 구성</figcaption></figure> <p><a href="/wiki/%EC%97%90%EC%9A%B0%ED%81%B4%EB%A0%88%EC%9D%B4%EB%8D%B0%EC%8A%A4" title="에우클레이데스">에우클레이데스</a>는 평면 기하에서의 삼각 부등식을 그림과 같은 구성을 사용하여 증명하였다.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> 삼각형 ABC에 대하여 일변 BC를 공유하는 이등변 삼각형을 또 하나의 등변 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">BD</span>의 아래쪽 변 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">AB</span>의 연장 위에 있도록 만든다. 그러면 모서리에 붙어서 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>β</i> > <i>α</i></span>를 말할 수 있기 때문에 변에 대해 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><span style="text-decoration:overline;">AD</span> > <span style="text-decoration:overline;">AC</span></span>도 (부등식을 <a href="/wiki/%EC%82%AC%EC%9D%B8_%EB%B2%95%EC%B9%99" title="사인 법칙">사인 법칙</a>에 근거해서 생각해보는 것도 가능)말할 수 있다. 그러나 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><span style="text-decoration:overline;">AD</span> = <span style="text-decoration:overline;">AB</span> + <span style="text-decoration:overline;">BD</span> = <span style="text-decoration:overline;">AB</span> + <span style="text-decoration:overline;">BC</span></span>이므로 변의 합에 대해 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><span style="text-decoration:overline;">AB</span> + <span style="text-decoration:overline;">BC</span> > <span style="text-decoration:overline;">AC</span></span>가 된다는 것이 《<a href="/wiki/%EC%97%90%EC%9A%B0%ED%81%B4%EB%A0%88%EC%9D%B4%EB%8D%B0%EC%8A%A4%EC%9D%98_%EC%9B%90%EB%A1%A0" title="에우클레이데스의 원론">에우클레이데스의 원론</a>》 제1권의 20번째 명제에 적혀 있다.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="절선_부등식"><span id=".EC.A0.88.EC.84.A0_.EB.B6.80.EB.93.B1.EC.8B.9D"></span>절선 부등식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=2" title="부분 편집: 절선 부등식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>삼각 부등식은 <a href="/wiki/%EC%88%98%ED%95%99%EC%A0%81_%EA%B7%80%EB%82%A9%EB%B2%95" title="수학적 귀납법">수학적 귀납법</a>을 통해 임의의 절선에 관한 명제로 확장할 수 있다. 즉 그러한 절선의 모든 변 길이의 합은 그 절선의 두 끝점을 직선으로 묶은 길이보다 작아지는 것은 아니다. 특히 그 귀결은 다각형의 어떤 길이의 변도 나머지 모든 변의 길이의 합보다 반드시 작다는 것을 말할 수 있다. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Arclength.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/300px-Arclength.svg.png" decoding="async" width="300" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/450px-Arclength.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/600px-Arclength.svg.png 2x" data-file-width="582" data-file-height="152" /></a><figcaption>곡선의 호 길이는 절선 근사 길이의 상한으로 정의된다.</figcaption></figure> <p>이와 같이 절선에 대해 일반화하면 유클리드 기하학에서 두 점 사이를 연결하는 최단 곡선이 직선임을 나타낼 수 있다. </p><p>두 점 사이를 연결하는 절선이 그 두 점 사이를 연결하는 선분보다 짧아지지 않는다는 점에서 곡선의 호 길이가 그 곡선의 양 끝점 사이의 거리보다 짧아지지 않는다는 것을 따른다. 실제로 정의에 의해 곡선의 호 길이는 그것을 근사하는 절선의 길이 상한으로, 절선에 대한 결과는 끝점 간을 연결하는 선분이 모든 절선 근사 중에서 가장 짧다는 것이었다. 곡선의 호 길이는 임의의 절선 근사 길이 이상이기 때문에 곡선 그 자체가 직선경로보다 짧아질 수는 없다.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="고차원_단체_부등식"><span id=".EA.B3.A0.EC.B0.A8.EC.9B.90_.EB.8B.A8.EC.B2.B4_.EB.B6.80.EB.93.B1.EC.8B.9D"></span>고차원 단체 부등식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=3" title="부분 편집: 고차원 단체 부등식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>삼각 부등식을 보다 고차원으로 일반화한 것으로서 유클리드 공간 내의 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">n</span>-차원 <a href="/wiki/%EB%8B%A8%EC%B2%B4_(%EC%88%98%ED%95%99)" title="단체 (수학)">단체</a>의 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>n</i> − 1</span>차원 면의 초부피는 그 이외의 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">n</span>개의 <a href="/wiki/%EB%A9%B4_(%EA%B8%B0%ED%95%98%ED%95%99)" title="면 (기하학)">면</a> 초부피의 합 이하이다. 특히 <a href="/wiki/%EC%82%AC%EB%A9%B4%EC%B2%B4" title="사면체">사면체</a>의 한 삼각형 면의 면적은 다른 3개의 면의 전체 면적의 합 이하가 된다. </p> <div class="mw-heading mw-heading2"><h2 id="노름_벡터_공간"><span id=".EB.85.B8.EB.A6.84_.EB.B2.A1.ED.84.B0_.EA.B3.B5.EA.B0.84"></span>노름 벡터 공간</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=4" title="부분 편집: 노름 벡터 공간"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Vector-triangle-inequality.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Vector-triangle-inequality.svg/300px-Vector-triangle-inequality.svg.png" decoding="async" width="300" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Vector-triangle-inequality.svg/450px-Vector-triangle-inequality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Vector-triangle-inequality.svg/600px-Vector-triangle-inequality.svg.png 2x" data-file-width="625" data-file-height="269" /></a><figcaption>벡터의 노름에 대한 삼각 부등식</figcaption></figure> <p><a href="/wiki/%EB%85%B8%EB%A6%84_%EA%B3%B5%EA%B0%84" title="노름 공간">노름 공간</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">V</span>에 대해 노름을 정의하는 성질 가운데 하나가 삼각 부등식 <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+y\|\leq \|x\|+\|y\|\quad (\forall \,x,y\in V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x+y\|\leq \|x\|+\|y\|\quad (\forall \,x,y\in V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f349068695c554b7d17b2a5937847a862a15412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.683ex; height:2.843ex;" alt="{\displaystyle \|x+y\|\leq \|x\|+\|y\|\quad (\forall \,x,y\in V)}"></span>이다. 즉 2개의 벡터의 합의 노름은 그 2개의 벡터 각각의 길이의 합으로 억제된다. 이를 <a href="/wiki/%EC%A4%80%EA%B0%80%EB%B2%95%EC%84%B1" title="준가법성">준가법성</a>이라고 부르기도 한다. 노름으로 사용할 것으로 기대되는 임의의 함수는 이 요건을 만족해야 한다.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>노름 공간이 <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간">유클리드 공간</a> 또는 보다 일반적인 엄격한 볼록 공간이라면 <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+y\|=\|x\|+\|y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x+y\|=\|x\|+\|y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ead3bc060a4146a3b4b997b6ccbe7216665e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.724ex; height:2.843ex;" alt="{\displaystyle \|x+y\|=\|x\|+\|y\|}"></span>가 되기 위한 필요 충분 조건은 3개의 점 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>x</i>, <i>y</i>, <i>x</i> + <i>y</i></span>가 형성하는 삼각형이 퇴화되어 있는 것, 즉 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x, y</span>가 동일한 반직선 상에 있는 것이다. 등식으로 표기하면 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>x</i> = 0</span> 또는 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>y</i> = 0</span> 또는 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>x</i> = <i>αy</i> (∃<i>α</i> > 0</span>이 된다. 이러한 성질은 엄격한 볼록 노름 공간(예를 들어 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">L<sub>p</sub></span>-공간 (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">1 < <i>p</i> < ∞</span> 등)을 특정짓는다. 그러나 이것이 성립되지 않는 노름 공간도 존재한다.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>주해 2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="거리_공간"><span id=".EA.B1.B0.EB.A6.AC_.EA.B3.B5.EA.B0.84"></span>거리 공간</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=5" title="부분 편집: 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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">M</span>의 거리 함수를 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">d</span>라고 하면 삼각 부등식 <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,z)\leq d(x,y)+d(y,z)\quad (\forall x,y,z\in M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z)\quad (\forall x,y,z\in M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9772d9a8737e8f5e785e41e1b049f7d2ff2eeea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.611ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)\quad (\forall x,y,z\in M)}"></span>는 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%ED%95%A8%EC%88%98" title="거리 함수">거리 함수</a>의 정의 요건 가운데 하나가 된다. 즉 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x</span>에서 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">z</span>까지의 거리는 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x</span>에서 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">y</span>까지의 거리와 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">y</span>에서 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">z</span>까지의 거리의 합으로서 위쪽에서부터 억제된다. </p><p>삼각 부등식은 거리 공간상의 흥미의 대부분을 차지하는 수렴과 관련되어 있다. 이는 거리 함수의 나머지 요건이 비교적 단순한 것에 기인한다. 예를 들어 거리 공간에서의 임의의 수렴 열이 <a href="/wiki/%EC%BD%94%EC%8B%9C_%EC%97%B4" title="코시 열">코시 열</a>이라는 사실은 삼각 부등식으로부터의 직접적인 귀결이다. 무엇이면 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x<sub>n</sub></span> 및 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x<sub>m</sub></span>을 (거리 공간에서의 수렴의 정의에 있는 경로에서의) 임의의 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>ε</i> > 0</span>에 대해 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>d</i>(<i>x<sub>n</sub></i>, <i>x</i>) < <i>ε</i>/2</span> 및 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>d</i>(<i>x<sub>m</sub></i>, <i>x</i>) < <i>ε</i>/2</span>가 되도록 취하면 삼각 부등식에 의해 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>d</i>(<i>x<sub>n</sub></i>, <i>x<sub>m</sub></i>) ≤ <i>d</i>(<i>x<sub>n</sub></i>, <i>x</i>) + <i>d</i>(<i>x<sub>m</sub></i>, <i>x</i>) < <i>ε</i>/2 + <i>ε</i>/2 = <i>ε</i></span>이 되고 점렬 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>x<sub>n</sub></i>}</span>은 정의에 따라 코시 열이 된다. </p><p>노름 공간을 노름이 유도하는 거리 함수 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>d</i>(<i>x</i>, <i>y</i>) ≔ ‖<i>x</i> − <i>y</i>‖</span>에서 거리 공간으로 보고 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>x</i> − <i>y</i></span>는 시작점 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">y</span>로부터 종점 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x</span>로 묶은 벡터로 해석할 때 이 공간의 거리 공간으로서의 삼각 부등식은 앞에서 언급한 노름 공간의 경우의 삼각 부등식으로 귀착된다. </p> <div class="mw-heading mw-heading2"><h2 id="역삼각_부등식"><span id=".EC.97.AD.EC.82.BC.EA.B0.81_.EB.B6.80.EB.93.B1.EC.8B.9D"></span>역삼각 부등식</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=6" title="부분 편집: 역삼각 부등식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>삼각 부등식이 위로부터의 평가인데 반해 아래로부터의 평가를 주는 "역방향 삼각 부등식"(reverse triangle inequality)은 삼각 부등식으로부터의 초등적인 귀결로서 얻는다. 그것은 평면 기하의 말로 말하면 "삼각형의 임의의 변은 그 외의 두 변의 차이보다 크다."라고 할 수 있다.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>노름 공간인 경우에는 <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 {\bigg |}\|x\|-\|y\|{\bigg |}\leq \|x-y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg |}\|x\|-\|y\|{\bigg |}\leq \|x-y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7d21767647ecc1b77500af062f696992940337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.018ex; height:5.176ex;" alt="{\displaystyle {\bigg |}\|x\|-\|y\|{\bigg |}\leq \|x-y\|}"></span>, 또는 거리 공간인 경우에는 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">|<i>d</i>(<i>y</i>, <i>x</i>) − <i>d</i>(<i>x</i>, <i>z</i>)| ≤ <i>d</i>(<i>y</i>, <i>z</i>)</span>이 된다. 이는 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">‖ • ‖</span>이나 거리 함수 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">d(x, •)</span>가 <span class="nowrap">립시츠 정수 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml">1</span></span>의 <a href="/wiki/%EB%A6%BD%EC%8B%9C%EC%B8%A0_%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="립시츠 연속 함수">립시츠 연속 함수</a>가 됨을 나타낸다. 따라서 특정한 <a href="/wiki/%EA%B7%A0%EB%93%B1_%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="균등 연속 함수">균등 연속 함수</a>이다. </p><p>역삼각 부등식은 일반적인 삼각 부등식을 사용하여 증명할 수 있다. </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\|=\|(x-y)+y\|\leq \|x-y\|+\|y\|\implies \|x\|-\|y\|\leq \|x-y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|=\|(x-y)+y\|\leq \|x-y\|+\|y\|\implies \|x\|-\|y\|\leq \|x-y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b39be1e371effa807db1a9e1f233fb3b1e5aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.389ex; height:2.843ex;" alt="{\displaystyle \|x\|=\|(x-y)+y\|\leq \|x-y\|+\|y\|\implies \|x\|-\|y\|\leq \|x-y\|}"></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 \|y\|=\|(y-x)+x\|\leq \|y-x\|+\|x\|\implies \|x\|-\|y\|\geq -\|x-y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|y\|=\|(y-x)+x\|\leq \|y-x\|+\|x\|\implies \|x\|-\|y\|\geq -\|x-y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83720a42c50b118502fd0f5f9b2a3d7b50d877bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.371ex; height:2.843ex;" alt="{\displaystyle \|y\|=\|(y-x)+x\|\leq \|y-x\|+\|x\|\implies \|x\|-\|y\|\geq -\|x-y\|}"></span></dd></dl> <p>이 점에 주의하면 <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-y\|\leq \|x\|-\|y\|\leq \|x-y\|\implies {\bigl |}\,\|x\|-\|y\|\,{\bigr |}\leq \|x-y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\|x-y\|\leq \|x\|-\|y\|\leq \|x-y\|\implies {\bigl |}\,\|x\|-\|y\|\,{\bigr |}\leq \|x-y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79cb1ecfb0122f0e8c1af429bdf32049deb6ecd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.46ex; height:3.176ex;" alt="{\displaystyle -\|x-y\|\leq \|x\|-\|y\|\leq \|x-y\|\implies {\bigl |}\,\|x\|-\|y\|\,{\bigr |}\leq \|x-y\|}"></span>이 된다. </p> <div class="mw-heading mw-heading2"><h2 id="민코프스키_공간에서의_부등호_반전"><span id=".EB.AF.BC.EC.BD.94.ED.94.84.EC.8A.A4.ED.82.A4_.EA.B3.B5.EA.B0.84.EC.97.90.EC.84.9C.EC.9D.98_.EB.B6.80.EB.93.B1.ED.98.B8_.EB.B0.98.EC.A0.84"></span>민코프스키 공간에서의 부등호 반전</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=7" title="부분 편집: 민코프스키 공간에서의 부등호 반전"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>민코프스키 공간에서 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x, y</span>가 함께 미래의 광원 뿔 안에 있는 시간적 벡터라면 삼각 부등식은 역방향의 평가 <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+y\|\geq \|x\|+\|y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x+y\|\geq \|x\|+\|y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901d56b01f4849647cbf248f0644cd9daeba2755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.724ex; height:2.843ex;" alt="{\displaystyle \|x+y\|\geq \|x\|+\|y\|}"></span>를 받게 된다. 이러한 부등식의 물리학적 예가 <a href="/w/index.php?title=%ED%8A%B9%EC%88%98_%EC%83%81%EB%8C%80%EB%A1%A0%EC%9D%B4%EB%A1%A0&action=edit&redlink=1" class="new" title="특수 상대론이론 (없는 문서)">특수 상대론이론</a>에서의 <a href="/wiki/%EC%8C%8D%EB%91%A5%EC%9D%B4_%EC%97%AD%EC%84%A4" title="쌍둥이 역설">쌍둥이 역설</a>이다. 2개의 벡터가 모두 과거의 광원 뿔 안에 있는 경우나 적어도 한 쪽이 눌 벡터인 경우에도 마찬가지로 이러한 역방향의 부등호를 갖는 삼각 부등식이 성립한다. 이 결과는 임의의 자연수 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">n</span>에 대한 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>n</i>+1</span>차원에서 성립한다. </p><p><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x, y</span>가 모두 공간적 벡터의 경우는 일반적인 삼각 부등식이 만족된다. </p> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=8" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%A4%80%EA%B0%80%EB%B2%95%EC%84%B1" title="준가법성">준가법성</a> (準加法性, Subadditivity)</li> <li><a href="/wiki/%EB%AF%BC%EC%BD%94%ED%94%84%EC%8A%A4%ED%82%A4_%EB%B6%80%EB%93%B1%EC%8B%9D" title="민코프스키 부등식">민코프스키 부등식</a></li></ul> <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%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=9" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="내용주"><span id=".EB.82.B4.EC.9A.A9.EC.A3.BC"></span>내용주</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=10" title="부분 편집: 내용주"><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-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">z</span>가 최대 변이 아닐 때는 오히려 분명해진다. (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>z</i> ≤ max(<i>x</i>, <i>y</i>) < <i>x</i> + <i>y</i>)</span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">예를 들어 평면에 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>L</i><sub>1</sub></span>-노름(즉 <a href="/wiki/%EB%A7%A8%ED%95%B4%ED%8A%BC_%EA%B1%B0%EB%A6%AC" title="맨해튼 거리">맨해튼 거리</a>)을 넣고 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>x</i> = (1, 0)</span> 및 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>y</i> = (0, 1)</span>을 취하면 3개의 점 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">x</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml" style="font-style: italic;">y</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25030363"><span class="texhtml"><i>x</i> + <i>y</i></span>를 형성하는 삼각형은 퇴화하지 않고 <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+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c296d730dfdb7fd54857f18177a6805a910f7018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.428ex; height:2.843ex;" alt="{\displaystyle \|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|}"></span>를 만족한다.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="참고주"><span id=".EC.B0.B8.EA.B3.A0.EC.A3.BC"></span>참고주</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=11" title="부분 편집: 참고주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r35556958"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TriangleInequality.html">“Triangle Inequality”</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=Triangle+Inequality&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTriangleInequality.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" 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 book">Mohamed A. Khamsi; William A. Kirk (2001년). 〈§1.4 The triangle inequality in ℝ<sup>n</sup>〉. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4qXbEpAK5eUC&pg=PA8">《An introduction to metric spaces and fixed point theory》</a>. Wiley-IEEE. <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%B0/0-471-41825-0" title="특수:책찾기/0-471-41825-0"><bdi>0-471-41825-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A71.4+The+triangle+inequality+in+%E2%84%9D%3Csup%3En%3C%2Fsup%3E&rft.btitle=An+introduction+to+metric+spaces+and+fixed+point+theory&rft.pub=Wiley-IEEE&rft.date=2001&rft.isbn=0-471-41825-0&rft.au=Mohamed+A.+Khamsi&rft.au=William+A.+Kirk&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4qXbEpAK5eUC%26pg%3DPA8&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" 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">Oliver Brock; Jeff Trinkle; Fabio Ramos (2009년). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fvCaQfBQ7qEC&pg=PA195">《Robotics: Science and Systems IV》</a>. MIT Press. 195쪽. <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%B0/978-0-262-51309-8" title="특수:책찾기/978-0-262-51309-8"><bdi>978-0-262-51309-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robotics%3A+Science+and+Systems+IV&rft.pages=195&rft.pub=MIT+Press&rft.date=2009&rft.isbn=978-0-262-51309-8&rft.au=Oliver+Brock&rft.au=Jeff+Trinkle&rft.au=Fabio+Ramos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfvCaQfBQ7qEC%26pg%3DPA195&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><cite class="citation book">Arlan Ramsay; Robert D. Richtmyer (1995년). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams">《Introduction to hyperbolic geometry》</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontohy0000rams/page/17">17</a>쪽. <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%B0/0-387-94339-0" title="특수:책찾기/0-387-94339-0"><bdi>0-387-94339-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+hyperbolic+geometry&rft.pages=17&rft.date=1995&rft.isbn=0-387-94339-0&rft.au=Arlan+Ramsay&rft.au=Robert+D.+Richtmyer&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontohy0000rams&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">지원되지 않는 변수 무시됨: <code style="color:inherit; border:inherit; padding:inherit;">|풀판사=</code> (<a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%9D%B8%EC%9A%A9_%EC%98%A4%EB%A5%98_%EB%8F%84%EC%9B%80%EB%A7%90#parameter_ignored" title="위키백과:인용 오류 도움말">도움말</a>)</span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><cite class="citation book">Harold R. Jacobs (2003년). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XhQRgZRDDq0C&pg=PA201">《Geometry: seeing, doing, understanding》</a> 3판. Macmillan. 201쪽. <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%B0/0-7167-4361-2" title="특수:책찾기/0-7167-4361-2"><bdi>0-7167-4361-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+seeing%2C+doing%2C+understanding&rft.pages=201&rft.edition=3&rft.pub=Macmillan&rft.date=2003&rft.isbn=0-7167-4361-2&rft.au=Harold+R.+Jacobs&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXhQRgZRDDq0C%26pg%3DPA201&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><cite class="citation web">David E. Joyce (1997년). <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI20.html">“Euclid's elements, Book 1, Proposition 20”</a>. Dept. Math and Computer Science, Clark University<span class="reference-accessdate">. 2010년 6월 25일에 확인함</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Euclid%27s+elements%2C+Book+1%2C+Proposition+20&rft.pub=Dept.+Math+and+Computer+Science%2C+Clark+University&rft.date=1997&rft.au=David+E.+Joyce&rft_id=http%3A%2F%2Faleph0.clarku.edu%2F~djoyce%2Fjava%2Felements%2FbookI%2FpropI20.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><cite class="citation book">John Stillwell (1997년). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4elkHwVS0eUC&pg=PA95">《Numbers and Geometry》</a>. Springer. <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%B0/978-0-387-98289-2" title="특수:책찾기/978-0-387-98289-2"><bdi>978-0-387-98289-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numbers+and+Geometry&rft.pub=Springer&rft.date=1997&rft.isbn=978-0-387-98289-2&rft.au=John+Stillwell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4elkHwVS0eUC%26pg%3DPA95&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span> p. 95.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><cite class="citation book">Rainer Kress (1988년). 〈§3.1: Normed spaces〉. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e7ZmHRIxum0C&pg=PA26">《Numerical analysis》</a>. Springer. 26쪽. <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%B0/0-387-98408-9" title="특수:책찾기/0-387-98408-9"><bdi>0-387-98408-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A73.1%3A+Normed+spaces&rft.btitle=Numerical+analysis&rft.pages=26&rft.pub=Springer&rft.date=1988&rft.isbn=0-387-98408-9&rft.au=Rainer+Kress&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De7ZmHRIxum0C%26pg%3DPA26&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text"><cite class="citation book">Anonymous (1854년). 〈Exercise I. to proposition XIX〉. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lTACAAAAQAAJ&pg=PA196">《The popular educator; fourth volume》</a>. Ludgate Hill, London: John Cassell. 196쪽.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Exercise+I.+to+proposition+XIX&rft.btitle=The+popular+educator%3B+fourth+volume&rft.place=Ludgate+Hill%2C+London&rft.pages=196&rft.pub=John+Cassell&rft.date=1854&rft.au=Anonymous&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlTACAAAAQAAJ%26pg%3DPA196&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" 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%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D&action=edit&section=12" 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://proofwiki.org/wiki/Triangle_inequality">“Triangle inequality”</a>. 《ProofWiki》 (영어).</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=Triangle+inequality&rft_id=http%3A%2F%2Fproofwiki.org%2Fwiki%2FTriangle_inequality&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%82%BC%EA%B0%81+%EB%B6%80%EB%93%B1%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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