Real coordinate space

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class="vector-toc-numb">4.3</span> <span>Convexity</span> </div> </a> <ul id="toc-Convexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Euclidean space</span> </div> </a> <ul id="toc-Euclidean_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_algebraic_and_differential_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_algebraic_and_differential_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>In algebraic and differential geometry</span> </div> </a> <ul id="toc-In_algebraic_and_differential_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_appearances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_appearances"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Other appearances</span> </div> </a> <ul id="toc-Other_appearances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polytopes_in_Rn" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polytopes_in_Rn"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Polytopes in R<sup><i>n</i></sup></span> </div> </a> <ul id="toc-Polytopes_in_Rn-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Topological_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topological_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Topological properties</span> </div> </a> <ul id="toc-Topological_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-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>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-n_≤_1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#n_≤_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span><i>n</i> ≤ 1</span> </div> </a> <ul id="toc-n_≤_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-n_=_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#n_=_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span><i>n</i> = 2</span> </div> </a> <ul id="toc-n_=_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-n_=_3" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#n_=_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span><i>n</i> = 3</span> </div> </a> <ul id="toc-n_=_3-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-n_=_4" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#n_=_4"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span><i>n</i> = 4</span> </div> </a> <ul id="toc-n_=_4-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Norms_on_Rn" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Norms_on_Rn"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Norms on <span><b>R</b><sup><i>n</i></sup></span></span> </div> </a> <ul id="toc-Norms_on_Rn-sublist" class="vector-toc-list"> </ul> 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message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cartesian-coordinate-system_v2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Cartesian-coordinate-system_v2.svg/220px-Cartesian-coordinate-system_v2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Cartesian-coordinate-system_v2.svg/330px-Cartesian-coordinate-system_v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Cartesian-coordinate-system_v2.svg/440px-Cartesian-coordinate-system_v2.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption><a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> identify points of the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> with pairs of real numbers</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>real coordinate space</b> or <b>real coordinate <i>n</i>-space</b>, of <a href="/wiki/Dimension" title="Dimension">dimension</a> <span class="texhtml mvar" style="font-style:italic;">n</span>, denoted <span class="texhtml"><b>R</b><sup><span class="texhtml mvar" style="font-style:italic;">n</span></sup></span> or <span class="nowrap"><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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span></span>, is the set of all ordered <a href="/wiki/Tuple" title="Tuple"><span class="texhtml mvar" style="font-style:italic;">n</span>-tuples</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a>, that is the set of all sequences of <span class="texhtml mvar" style="font-style:italic;">n</span> real numbers, also known as <i><a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinate vectors</a></i>. Special cases are called the <i><a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a></i> <span class="texhtml"><b>R</b><sup>1</sup></span>, the <i>real coordinate plane</i> <span class="texhtml"><b>R</b><sup>2</sup></span>, and the <i>real coordinate three-dimensional space</i> <span class="texhtml"><b>R</b><sup>3</sup></span>. With component-wise addition and scalar multiplication, it is a <a href="/wiki/Real_vector_space" class="mw-redirect" title="Real vector space">real vector space</a>. </p><p>The <a href="/wiki/Coordinate_(vector_space)" class="mw-redirect" title="Coordinate (vector space)">coordinates</a> over any <a href="/wiki/Basis_(vector_space)" class="mw-redirect" title="Basis (vector space)">basis</a> of the elements of a real vector space form a <i>real coordinate space</i> of the same dimension as that of the vector space. Similarly, the <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of the points of a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml"><b>E</b><sup>n</sup></span> (<a href="/wiki/Euclidean_line" class="mw-redirect" title="Euclidean line">Euclidean line</a>, <span class="texhtml"><b>E</b></span>; <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, <span class="texhtml"><b>E</b><sup>2</sup></span>; <a href="/wiki/Euclidean_three-dimensional_space" class="mw-redirect" title="Euclidean three-dimensional space">Euclidean three-dimensional space</a>, <span class="texhtml"><b>E</b><sup>3</sup></span>) form a <i>real coordinate space</i> of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>These <a href="/wiki/One_to_one_correspondence" class="mw-redirect" title="One to one correspondence">one to one correspondences</a> between vectors, points and coordinate vectors explain the names of <i>coordinate space</i> and <i>coordinate vector</i>. It allows using <a href="/wiki/Geometric" class="mw-redirect" title="Geometric">geometric</a> terms and methods for studying real coordinate spaces, and, conversely, to use methods of <a href="/wiki/Calculus" title="Calculus">calculus</a> in geometry. This approach of geometry was introduced by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_structures">Definition and structures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=1" title="Edit section: Definition and structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="texhtml mvar" style="font-style:italic;">n</span>, the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> consists of all <span class="texhtml mvar" style="font-style:italic;">n</span>-<a href="/wiki/Tuple" title="Tuple">tuples</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a> (<span class="texhtml"><b>R</b></span>). It is called the "<span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional real space" or the "real <span class="texhtml mvar" style="font-style:italic;">n</span>-space". </p><p>An element of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is thus a <span class="texhtml mvar" style="font-style:italic;">n</span>-tuple, and is written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\ldots ,x_{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"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d8b3ec62633086002489cad8df4214e9585880" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.337ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}"></span> where each <span class="texhtml"><i>x</i><sub><i>i</i></sub></span> is a real number. So, in <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>, the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of a <a href="/wiki/Function_of_several_real_variables" title="Function of several real variables">function of several real variables</a> and the codomain of a real <a href="/wiki/Vector_valued_function" class="mw-redirect" title="Vector valued function">vector valued function</a> are <a href="/wiki/Subset" title="Subset">subsets</a> of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> for some <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>The real <span class="texhtml mvar" style="font-style:italic;">n</span>-space has several further properties, notably: </p> <ul><li>With <a href="/wiki/Componentwise_operation" class="mw-redirect" title="Componentwise operation">componentwise</a> addition and <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>, it is a <a href="/wiki/Real_vector_space" class="mw-redirect" title="Real vector space">real vector space</a>. Every <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional real vector space is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to it.</li> <li>With the <a href="/wiki/Dot_product" title="Dot product">dot product</a> (sum of the term by term product of the components), it is an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>. Every <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional real inner product space is isomorphic to it.</li> <li>As every inner product space, it is a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, and a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a>.</li> <li>It is a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> and a real <a href="/wiki/Affine_space" title="Affine space">affine space</a>, and every Euclidean or affine space is isomorphic to it.</li> <li>It is an <a href="/wiki/Analytic_manifold" title="Analytic manifold">analytic manifold</a>, and can be considered as the prototype of all <a href="/wiki/Manifold" title="Manifold">manifolds</a>, as, by definition, a manifold is, near each point, isomorphic to an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>.</li> <li>It is an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a>, and every <a href="/wiki/Real_algebraic_variety" class="mw-redirect" title="Real algebraic variety">real algebraic variety</a> is a subset of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>.</li></ul> <p>These properties and structures of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> make it fundamental in almost all areas of mathematics and their application domains, such as <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, and many parts of <a href="/wiki/Physics" title="Physics">physics</a>. </p> <div class="mw-heading mw-heading2"><h2 id="The_domain_of_a_function_of_several_variables">The domain of a function of several variables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=2" title="Edit section: The domain of a function of several variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.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}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> and <a href="/wiki/Real_multivariable_function" class="mw-redirect" title="Real multivariable function">Real multivariable function</a></div> <p>Any function <span class="texhtml"><i>f</i>(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub>)</span> of <span class="texhtml mvar" style="font-style:italic;">n</span> real variables can be considered as a function on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> (that is, with <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> as its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>). The use of the real <span class="texhtml mvar" style="font-style:italic;">n</span>-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for <span class="texhtml"><i>n</i> = 2</span>, a <a href="/wiki/Function_composition" title="Function composition">function composition</a> of the following form: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)=f(g_{1}(t),g_{2}(t)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(t)=f(g_{1}(t),g_{2}(t)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a8ab031073ca734f0701e3ddc39d1ff7a6a76b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.881ex; height:2.843ex;" alt="{\displaystyle F(t)=f(g_{1}(t),g_{2}(t)),}"></span> where functions <span class="texhtml"><i>g</i><sub>1</sub></span> and <span class="texhtml"><i>g</i><sub>2</sub></span> are <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>. If </p> <ul><li><span class="texhtml">∀<i>x</i><sub>1</sub> ∈ <b>R</b> : <i>f</i>(<i>x</i><sub>1</sub>, ·)</span> is continuous (by <span class="texhtml"><i>x</i><sub>2</sub></span>)</li> <li><span class="texhtml">∀<i>x</i><sub>2</sub> ∈ <b>R</b> : <i>f</i>(·, <i>x</i><sub>2</sub>)</span> is continuous (by <span class="texhtml"><i>x</i><sub>1</sub></span>)</li></ul> <p>then <span class="texhtml mvar" style="font-style:italic;">F</span> is not necessarily continuous. Continuity is a stronger condition: the continuity of <span class="texhtml mvar" style="font-style:italic;">f</span> in the natural <span class="texhtml"><b>R</b><sup>2</sup></span> topology (<a href="#Topological_properties">discussed below</a>), also called <i>multivariable continuity</i>, which is sufficient for continuity of the composition <span class="texhtml mvar" style="font-style:italic;">F</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Vector_space">Vector space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=3" title="Edit section: Vector space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The coordinate space <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> forms an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional <a href="/wiki/Vector_space" title="Vector space">vector space</a> over the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of real numbers with the addition of the structure of <a href="/wiki/Linearity" title="Linearity">linearity</a>, and is often still denoted <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. The operations on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> as a vector space are typically defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} +\mathbf {y} =(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} +\mathbf {y} =(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f74339e62200beb94e659a50b661f04af563dc0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.364ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} +\mathbf {y} =(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \mathbf {x} =(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \mathbf {x} =(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2fa4a171d8312e430665a169f1774bdb22d159b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.444ex; height:2.843ex;" alt="{\displaystyle \alpha \mathbf {x} =(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n}).}"></span> The <a href="/wiki/Additive_identity" title="Additive identity">zero vector</a> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} =(0,0,\ldots ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} =(0,0,\ldots ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae1214c4abb0e3809bca46d99f9753b92cf3563" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.944ex; height:2.843ex;" alt="{\displaystyle \mathbf {0} =(0,0,\ldots ,0)}"></span> and the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> of the vector <span class="texhtml"><b>x</b></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathbf {x} =(-x_{1},-x_{2},\ldots ,-x_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathbf {x} =(-x_{1},-x_{2},\ldots ,-x_{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93c8f6301bc77d3e0b3a59220502937da0c21d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.726ex; height:2.843ex;" alt="{\displaystyle -\mathbf {x} =(-x_{1},-x_{2},\ldots ,-x_{n}).}"></span> </p><p>This structure is important because any <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional real vector space is isomorphic to the vector space <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_notation">Matrix notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=4" title="Edit section: Matrix notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix (mathematics)</a></div> <p>In standard <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> notation, each element of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is typically written as a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8d1334f9662f3c0d43bf5a454037e29daeb690" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:10.91ex; height:13.843ex;" alt="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}}"></span> and sometimes as a <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2e6274914356d365627e03e56bdf7429aa7ba8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.208ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}.}"></span> </p><p>The coordinate space <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> may then be interpreted as the space of all <span class="texhtml"><i>n</i> × 1</span> <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vectors</a>, or all <span class="texhtml">1 × <i>n</i></span> <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vectors</a> with the ordinary matrix operations of addition and <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>. </p><p><a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">Linear transformations</a> from <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> to <span class="texhtml"><b>R</b><sup><i>m</i></sup></span> may then be written as <span class="texhtml"><i>m</i> × <i>n</i></span> matrices which act on the elements of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> via <a href="/wiki/Left_and_right_(algebra)" title="Left and right (algebra)">left</a> multiplication (when the elements of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> are column vectors) and on elements of <span class="texhtml"><b>R</b><sup><i>m</i></sup></span> via right multiplication (when they are row vectors). The formula for left multiplication, a special case of <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>, is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A{\mathbf {x} })_{k}=\sum _{l=1}^{n}A_{kl}x_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A{\mathbf {x} })_{k}=\sum _{l=1}^{n}A_{kl}x_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f966fa616b145900c552aba8fc877118dcc8005c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.267ex; height:6.843ex;" alt="{\displaystyle (A{\mathbf {x} })_{k}=\sum _{l=1}^{n}A_{kl}x_{l}}"></span> </p><p><span class="anchor" id="continuity_of_linear_maps"></span>Any linear transformation is a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> (see <a href="#Topological_properties">below</a>). Also, a matrix defines an <a href="/wiki/Open_map" class="mw-redirect" title="Open map">open map</a> from <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> to <span class="texhtml"><b>R</b><sup><i>m</i></sup></span> if and only if the <a href="/wiki/Rank_(matrix_theory)" class="mw-redirect" title="Rank (matrix theory)">rank of the matrix</a> equals to <span class="texhtml mvar" style="font-style:italic;">m</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Standard_basis">Standard basis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=5" title="Edit section: Standard basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Standard_basis" title="Standard basis">Standard basis</a></div> <p>The coordinate space <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> comes with a standard basis: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {e} _{1}&=(1,0,\ldots ,0)\\\mathbf {e} _{2}&=(0,1,\ldots ,0)\\&{}\;\;\vdots \\\mathbf {e} _{n}&=(0,0,\ldots ,1)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {e} _{1}&=(1,0,\ldots ,0)\\\mathbf {e} _{2}&=(0,1,\ldots ,0)\\&{}\;\;\vdots \\\mathbf {e} _{n}&=(0,0,\ldots ,1)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3acc01aa66d6cbbf5a0d8644279e1469f902c76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:17.802ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {e} _{1}&=(1,0,\ldots ,0)\\\mathbf {e} _{2}&=(0,1,\ldots ,0)\\&{}\;\;\vdots \\\mathbf {e} _{n}&=(0,0,\ldots ,1)\end{aligned}}}"></span> </p><p>To see that this is a basis, note that an arbitrary vector in <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> can be written uniquely in the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {e} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {e} _{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d23572f2dce02e2d774ca470011847c22944e120" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.053ex; height:6.843ex;" alt="{\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {e} _{i}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Geometric_properties_and_uses">Geometric properties and uses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=6" title="Edit section: Geometric properties and uses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Orientation">Orientation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=7" title="Edit section: Orientation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fact that <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, unlike many other <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, constitute an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> yields an <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">orientation structure</a> on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. Any <a href="/wiki/Rank_(matrix_theory)" class="mw-redirect" title="Rank (matrix theory)">full-rank</a> linear map of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> to itself either preserves or reverses orientation of the space depending on the <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a> of the <a href="/wiki/Determinant" title="Determinant">determinant</a> of its matrix. If one <a href="/wiki/Permutation" title="Permutation">permutes</a> coordinates (or, in other words, elements of the basis), the resulting orientation will depend on the <a href="/wiki/Parity_of_a_permutation" title="Parity of a permutation">parity of the permutation</a>. </p><p><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphisms</a> of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> or <a href="/wiki/Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domains in it</a>, by their virtue to avoid zero <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a>, are also classified to orientation-preserving and orientation-reversing. It has important consequences for the theory of <a href="/wiki/Differential_form" title="Differential form">differential forms</a>, whose applications include <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>. </p><p>Another manifestation of this structure is that the <a href="/wiki/Point_reflection" title="Point reflection">point reflection</a> in <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> has different properties depending on <a href="/wiki/Even_and_odd_numbers" class="mw-redirect" title="Even and odd numbers">evenness of <span class="texhtml mvar" style="font-style:italic;">n</span></a>. For even <span class="texhtml mvar" style="font-style:italic;">n</span> it preserves orientation, while for odd <span class="texhtml mvar" style="font-style:italic;">n</span> it is reversed (see also <a href="/wiki/Improper_rotation" title="Improper rotation">improper rotation</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Affine_space">Affine space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=8" title="Edit section: Affine space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Affine_space" title="Affine space">Affine space</a></div> <p><span class="texhtml"><b>R</b><sup><i>n</i></sup></span> understood as an affine space is the same space, where <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> as a vector space <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a> by <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>. Conversely, a vector has to be understood as a "<a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">difference</a> between two points", usually illustrated by a directed <a href="/wiki/Line_segment" title="Line segment">line segment</a> connecting two points. The distinction says that there is no <a href="/wiki/Canonical_form" title="Canonical form">canonical</a> choice of where the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> should go in an affine <span class="texhtml mvar" style="font-style:italic;">n</span>-space, because it can be translated anywhere. </p> <div class="mw-heading mw-heading3"><h3 id="Convexity">Convexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=9" title="Edit section: Convexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2D-simplex.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/220px-2D-simplex.svg.png" decoding="async" width="220" height="314" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/330px-2D-simplex.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/440px-2D-simplex.svg.png 2x" data-file-width="96" data-file-height="137" /></a><figcaption>The <i>n</i>-simplex (see <a href="#Polytopes_in_Rn">below</a>) is the standard convex set, that maps to every polytope, and is the intersection of the standard <span class="texhtml">(<i>n</i> + 1)</span> affine hyperplane (standard affine space) and the standard <span class="texhtml">(<i>n</i> + 1)</span> orthant (standard cone).</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Convex_analysis" title="Convex analysis">Convex analysis</a></div> <p>In a real vector space, such as <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>, one can define a convex <a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">cone</a>, which contains all <i>non-negative</i> linear combinations of its vectors. Corresponding concept in an affine space is a <a href="/wiki/Convex_set" title="Convex set">convex set</a>, which allows only <a href="/wiki/Convex_combination" title="Convex combination">convex combinations</a> (non-negative linear combinations that sum to 1). </p><p>In the language of <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>, a vector space is an algebra over the universal vector space <span class="texhtml"><b>R</b><sup>∞</sup></span> of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal <a href="/wiki/Orthant" title="Orthant">orthant</a> (of finite sequences of nonnegative numbers), and a convex set is an algebra over the universal <a href="/wiki/Simplex" title="Simplex">simplex</a> (of finite sequences of nonnegative numbers summing to 1). This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates". </p><p>Another concept from convex analysis is a <a href="/wiki/Convex_function" title="Convex function">convex function</a> from <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> to real numbers, which is defined through an <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a> between its value on a convex combination of <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> and sum of values in those points with the same coefficients. </p> <div class="mw-heading mw-heading3"><h3 id="Euclidean_space">Euclidean space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=10" title="Edit section: Euclidean space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> and <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></div> <p>The <a href="/wiki/Dot_product" title="Dot product">dot product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \cdot \mathbf {y} =\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a71ba4838c7a2c83743b73f8deea6c5bd76b33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.813ex; height:6.843ex;" alt="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}}"></span> defines the <a href="/wiki/Normed_vector_space" title="Normed vector space">norm</a> <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><b>x</b></span>| = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><b>x</b> ⋅ <b>x</b></span></span></span> on the vector space <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. If every vector has its <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>, then for any pair of points the distance <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo 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 stretchy="false">)</mo> <mo>=</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <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> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be1e92c82a80285817604354275b99fb2bb5d4da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.516ex; height:7.509ex;" alt="{\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}"></span> is defined, providing a <a href="/wiki/Metric_space" title="Metric space">metric space</a> structure on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> in addition to its affine structure. </p><p>As for vector space structure, the dot product and Euclidean distance usually are assumed to exist in <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> without special explanations. However, the real <span class="texhtml mvar" style="font-style:italic;">n</span>-space and a Euclidean <span class="texhtml mvar" style="font-style:italic;">n</span>-space are distinct objects, strictly speaking. Any Euclidean <span class="texhtml mvar" style="font-style:italic;">n</span>-space has a <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> where the dot product and Euclidean distance have the form shown above, called <a href="/wiki/Renatus_Cartesius" class="mw-redirect" title="Renatus Cartesius"><i>Cartesian</i></a>. But there are <i>many</i> Cartesian coordinate systems on a Euclidean space. </p><p>Conversely, the above formula for the Euclidean metric defines the <i>standard</i> Euclidean structure on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>, but it is not the only possible one. Actually, any <a href="/wiki/Positive-definite_quadratic_form" class="mw-redirect" title="Positive-definite quadratic form">positive-definite quadratic form</a> <span class="texhtml mvar" style="font-style:italic;">q</span> defines its own "distance" <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>q</i>(<b>x</b> − <b>y</b>)</span></span></span>, but it is not very different from the Euclidean one in the sense that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists C_{1}>0,\ \exists C_{2}>0,\ \forall \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}:C_{1}d(\mathbf {x} ,\mathbf {y} )\leq {\sqrt {q(\mathbf {x} -\mathbf {y} )}}\leq C_{2}d(\mathbf {x} ,\mathbf {y} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <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>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>d</mi> <mo 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 stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>q</mi> <mo 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 stretchy="false">)</mo> </msqrt> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>d</mi> <mo 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 stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists C_{1}>0,\ \exists C_{2}>0,\ \forall \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}:C_{1}d(\mathbf {x} ,\mathbf {y} )\leq {\sqrt {q(\mathbf {x} -\mathbf {y} )}}\leq C_{2}d(\mathbf {x} ,\mathbf {y} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b433f99fd391eeb7f3e89516647c838a40d1a68" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:69.494ex; height:4.843ex;" alt="{\displaystyle \exists C_{1}>0,\ \exists C_{2}>0,\ \forall \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}:C_{1}d(\mathbf {x} ,\mathbf {y} )\leq {\sqrt {q(\mathbf {x} -\mathbf {y} )}}\leq C_{2}d(\mathbf {x} ,\mathbf {y} ).}"></span> Such a change of the metric preserves some of its properties, for example the property of being a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>. This also implies that any full-rank linear transformation of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>, or its <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a>, does not magnify distances more than by some fixed <span class="texhtml"><i>C</i><sub>2</sub></span>, and does not make distances smaller than <span class="texhtml">1 / <i>C</i><sub>1</sub></span> times, a fixed finite number times smaller.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (October 2014)">clarification needed</span></a></i>]</sup> </p><p>The aforementioned equivalence of metric functions remains valid if <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>q</i>(<b>x</b> − <b>y</b>)</span></span></span> is replaced with <span class="texhtml"><i>M</i>(<b>x</b> − <b>y</b>)</span>, where <span class="texhtml mvar" style="font-style:italic;">M</span> is any convex positive <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a> of degree 1, i.e. a <a href="/wiki/Normed_vector_space" title="Normed vector space">vector norm</a> (see <a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a> for useful examples). Because of this fact that any "natural" metric on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is not especially different from the Euclidean metric, <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is not always distinguished from a Euclidean <span class="texhtml"><i>n</i></span>-space even in professional mathematical works. </p> <div class="mw-heading mw-heading3"><h3 id="In_algebraic_and_differential_geometry">In algebraic and differential geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=11" title="Edit section: In algebraic and differential geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although the definition of a <a href="/wiki/Manifold" title="Manifold">manifold</a> does not require that its model space should be <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>, this choice is the most common, and almost exclusive one in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. </p><p>On the other hand, <a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding theorems</a> state that any real <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable <span class="texhtml mvar" style="font-style:italic;">m</span>-dimensional manifold</a> can be <a href="/wiki/Embedding" title="Embedding">embedded</a> into <span class="texhtml"><b>R</b><sup>2<i>m</i></sup></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_appearances">Other appearances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=12" title="Edit section: Other appearances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other structures considered on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> include the one of a <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a>, <a href="/wiki/Symplectic_structure" class="mw-redirect" title="Symplectic structure">symplectic structure</a> (even <span class="texhtml mvar" style="font-style:italic;">n</span>), and <a href="/wiki/Contact_structure" class="mw-redirect" title="Contact structure">contact structure</a> (odd <span class="texhtml mvar" style="font-style:italic;">n</span>). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates. </p><p><span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is also a real vector subspace of <span class="texhtml"><a href="/wiki/Complex_coordinate_space" title="Complex coordinate space"><b>C</b><sup><i>n</i></sup></a></span> which is invariant to <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>; see also <a href="/wiki/Complexification" title="Complexification">complexification</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Polytopes_in_Rn">Polytopes in R<sup><i>n</i></sup></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=13" title="Edit section: Polytopes in Rn"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Linear_programming" title="Linear programming">Linear programming</a> and <a href="/wiki/Convex_polytope" title="Convex polytope">Convex polytope</a></div> <p>There are three families of <a href="/wiki/Polytope" title="Polytope">polytopes</a> which have simple representations in <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> spaces, for any <span class="texhtml mvar" style="font-style:italic;">n</span>, and can be used to visualize any affine coordinate system in a real <span class="texhtml mvar" style="font-style:italic;">n</span>-space. Vertices of a <a href="/wiki/Hypercube" title="Hypercube">hypercube</a> have coordinates <span class="texhtml">(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub>)</span> where each <span class="texhtml mvar" style="font-style:italic;">x<sub>k</sub></span> takes on one of only two values, typically 0 or 1. However, any two numbers can be chosen instead of 0 and 1, for example <a href="/wiki/%E2%88%921_(number)" class="mw-redirect" title="−1 (number)">−1</a> and 1. An <span class="texhtml mvar" style="font-style:italic;">n</span>-hypercube can be thought of as the Cartesian product of <span class="texhtml mvar" style="font-style:italic;">n</span> identical <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a> (such as the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> <span class="texhtml">[0,1]</span>) on the real line. As an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional subset it can be described with a <a href="/wiki/System_of_inequalities" class="mw-redirect" title="System of inequalities">system of <span class="texhtml">2<i>n</i></span> inequalities</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}0\leq x_{1}\leq 1\\\vdots \\0\leq x_{n}\leq 1\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}0\leq x_{1}\leq 1\\\vdots \\0\leq x_{n}\leq 1\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90dc08227b0f40b60d3d6fe4af7aad4834065d55" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:11.821ex; height:10.509ex;" alt="{\displaystyle {\begin{matrix}0\leq x_{1}\leq 1\\\vdots \\0\leq x_{n}\leq 1\end{matrix}}}"></span> for <span class="texhtml">[0,1]</span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}|x_{1}|\leq 1\\\vdots \\|x_{n}|\leq 1\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}|x_{1}|\leq 1\\\vdots \\|x_{n}|\leq 1\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cedf8da0964776e9045eecb87d0f39311c5a6d8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:8.854ex; height:10.843ex;" alt="{\displaystyle {\begin{matrix}|x_{1}|\leq 1\\\vdots \\|x_{n}|\leq 1\end{matrix}}}"></span> for <span class="texhtml">[−1,1]</span>. </p> <div style="clear:left;" class=""></div> <p>Each vertex of the <a href="/wiki/Cross-polytope" title="Cross-polytope">cross-polytope</a> has, for some <span class="texhtml mvar" style="font-style:italic;">k</span>, the <span class="texhtml mvar" style="font-style:italic;">x<sub>k</sub></span> coordinate equal to <a href="/w/index.php?title=%C2%B11&action=edit&redlink=1" class="new" title="±1 (page does not exist)">±1</a> and all other coordinates equal to 0 (such that it is the <span class="texhtml mvar" style="font-style:italic;">k</span>th <a href="#Standard_basis">standard basis vector</a> up to <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a>). This is a <a href="/wiki/Dual_polytope" class="mw-redirect" title="Dual polytope">dual polytope</a> of hypercube. As an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional subset it can be described with a single inequality which uses the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> operation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}|x_{k}|\leq 1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}|x_{k}|\leq 1\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4fcf93ea8c62701b15bc7a7ffb84adaf70e749" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.749ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}|x_{k}|\leq 1\,,}"></span> but this can be expressed with a system of <span class="texhtml"><a href="/wiki/Power_of_two" title="Power of two">2<sup><i>n</i></sup></a></span> linear inequalities as well. </p><p>The third polytope with simply enumerable coordinates is the <a href="/wiki/Standard_simplex" class="mw-redirect" title="Standard simplex">standard simplex</a>, whose vertices are <span class="texhtml mvar" style="font-style:italic;">n</span> standard basis vectors and <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">the origin</a> <span class="texhtml">(0, 0, ..., 0)</span>. As an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional subset it is described with a system of <span class="texhtml"><i>n</i> + 1</span> linear inequalities: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}0\leq x_{1}\\\vdots \\0\leq x_{n}\\\sum \limits _{k=1}^{n}x_{k}\leq 1\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}0\leq x_{1}\\\vdots \\0\leq x_{n}\\\sum \limits _{k=1}^{n}x_{k}\leq 1\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca4ee965923a7da981732b6463bf54d827325c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:10.775ex; height:16.843ex;" alt="{\displaystyle {\begin{matrix}0\leq x_{1}\\\vdots \\0\leq x_{n}\\\sum \limits _{k=1}^{n}x_{k}\leq 1\end{matrix}}}"></span> Replacement of all "≤" with "<" gives interiors of these polytopes. </p> <div class="mw-heading mw-heading2"><h2 id="Topological_properties">Topological properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=14" title="Edit section: Topological properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Topology_(structure)" class="mw-redirect" title="Topology (structure)">topological structure</a> of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> (called <b>standard topology</b>, <b>Euclidean topology</b>, or <b>usual topology</b>) can be obtained not only <a href="#Definition_and_uses">from Cartesian product</a>. It is also identical to the <a href="/wiki/Natural_topology" title="Natural topology">natural topology</a> induced by <a href="#Euclidean_space">Euclidean metric discussed above</a>: a set is <a href="/wiki/Open_set" title="Open set">open</a> in the Euclidean topology <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it contains an <a href="/wiki/Open_ball" class="mw-redirect" title="Open ball">open ball</a> around each of its points. Also, <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is a <a href="/wiki/Linear_topological_space" class="mw-redirect" title="Linear topological space">linear topological space</a> (see <a href="#continuity_of_linear_maps">continuity of linear maps</a> above), and there is only one possible (non-trivial) topology compatible with its linear structure. As there are many open linear maps from <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> to itself which are not <a href="/wiki/Isometry" title="Isometry">isometries</a>, there can be many Euclidean structures on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> which correspond to the same topology. Actually, it does not depend much even on the linear structure: there are many non-linear <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a> (and other homeomorphisms) of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> onto itself, or its parts such as a Euclidean open ball or <a href="#Polytopes_in_Rn">the interior of a hypercube</a>). </p><p><span class="texhtml"><b>R</b><sup><i>n</i></sup></span> has the <a href="/wiki/Topological_dimension" class="mw-redirect" title="Topological dimension">topological dimension</a> <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>An important result on the topology of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>, that is far from superficial, is <a href="/wiki/L._E._J._Brouwer" title="L. E. J. Brouwer">Brouwer</a>'s <a href="/wiki/Invariance_of_domain" title="Invariance of domain">invariance of domain</a>. Any subset of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> (with its <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>) that is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to another open subset of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is itself open. An immediate consequence of this is that <span class="texhtml"><b>R</b><sup><i>m</i></sup></span> is not <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> to <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> if <span class="texhtml"><i>m</i> ≠ <i>n</i></span> – an intuitively "obvious" result which is nonetheless difficult to prove. </p><p>Despite the difference in topological dimension, and contrary to a naïve perception, it is possible to map a lesser-dimensional<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (April 2016)">clarification needed</span></a></i>]</sup> real space continuously and <a href="/wiki/Surjective_function" title="Surjective function">surjectively</a> onto <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. A continuous (although not smooth) <a href="/wiki/Space-filling_curve" title="Space-filling curve">space-filling curve</a> (an image of <span class="texhtml"><b>R</b><sup>1</sup></span>) is possible.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (April 2016)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=15" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" style="margin: 2ex 0 2ex 2em"> <tbody><tr> <td align="center"><span typeof="mw:File"><a href="/wiki/File:Real_0-space.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Real_0-space.svg/52px-Real_0-space.svg.png" decoding="async" width="52" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Real_0-space.svg/78px-Real_0-space.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Real_0-space.svg/104px-Real_0-space.svg.png 2x" data-file-width="104" data-file-height="40" /></a></span> </td></tr> <tr> <td style="font-size:80%"><a href="/wiki/Empty_matrix" class="mw-redirect" title="Empty matrix">Empty</a> column vector,<br />the only element of <span class="texhtml"><b>R</b><sup>0</sup></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="n_≤_1"><span id="n_.E2.89.A4_1"></span><i>n</i> ≤ 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=16" title="Edit section: n ≤ 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cases of <span class="texhtml">0 ≤ <i>n</i> ≤ 1</span> do not offer anything new: <span class="texhtml"><b>R</b><sup>1</sup></span> is the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, whereas <span class="texhtml"><b>R</b><sup>0</sup></span> (the space containing the empty column vector) is a <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singleton</a>, understood as a <a href="/wiki/Zero_vector_space" class="mw-redirect" title="Zero vector space">zero vector space</a>. However, it is useful to include these as <a href="/wiki/Triviality_(mathematics)" title="Triviality (mathematics)">trivial</a> cases of theories that describe different <span class="texhtml mvar" style="font-style:italic;">n</span>. </p> <div class="mw-heading mw-heading3"><h3 id="n_=_2"><span id="n_.3D_2"></span><i>n</i> = 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=17" title="Edit section: n = 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Real_2-space,_orthoplex.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Real_2-space%2C_orthoplex.svg/264px-Real_2-space%2C_orthoplex.svg.png" decoding="async" width="264" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Real_2-space%2C_orthoplex.svg/396px-Real_2-space%2C_orthoplex.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Real_2-space%2C_orthoplex.svg/528px-Real_2-space%2C_orthoplex.svg.png 2x" data-file-width="528" data-file-height="536" /></a><figcaption>Both hypercube and cross-polytope in <span class="texhtml"><b>R</b><sup>2</sup></span> are <a href="/wiki/Square" title="Square">squares</a>, but coordinates of vertices are arranged differently</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional space</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/SL2(R)" title="SL2(R)">SL2(R)</a></div> <p>The case of (<i>x,y</i>) where <i>x</i> and <i>y</i> are real numbers has been developed as the <a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a> <i>P</i>. Further structure has been attached with <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vectors</a> representing directed line segments in <i>P</i>. The plane has also been developed as the <a href="/wiki/Field_extension" title="Field extension">field extension</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 \mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"></span> by appending roots of X<sup>2</sup> + 1 = 0 to the real field <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 {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a067bb21dcf0642bdce48f05a55e218efab3b85e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.65ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} .}"></span> The root i acts on P as a <a href="/wiki/Quarter_turn" class="mw-redirect" title="Quarter turn">quarter turn</a> with counterclockwise orientation. This root generates the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</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 \{i,-1,-i,+1\}\equiv \mathbf {Z} /4\mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mi>i</mi> <mo>,</mo> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{i,-1,-i,+1\}\equiv \mathbf {Z} /4\mathbf {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388d1808fd7f8f0df8a5d98c1524d7f6499e4eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.472ex; height:2.843ex;" alt="{\displaystyle \{i,-1,-i,+1\}\equiv \mathbf {Z} /4\mathbf {Z} }"></span>. When (<i>x,y</i>) is written <i>x</i> + <i>y</i> i it is a <a href="/wiki/Complex_number" title="Complex number">complex number</a>. </p><p>Another <a href="/wiki/Group_action" title="Group action">group action</a> by <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 {Z} /2\mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} /2\mathbf {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/979d4bc367549003cfac5b5580c63462dd944e68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.593ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} /2\mathbf {Z} }"></span>, where the actor has been expressed as j, uses the line <i>y</i>=<i>x</i> for the <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> of flipping the plane (<i>x,y</i>) ↦ (<i>y,x</i>), an exchange of coordinates. In this case points of <i>P</i> are written <i>x</i> + <i>y</i> j and called <a href="/wiki/Split-complex_number" title="Split-complex number">split-complex numbers</a>. These numbers, with the coordinate-wise addition and multiplication according to <i>jj</i>=+1, form a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> that is not a field. </p><p>Another ring structure on <i>P</i> uses a <a href="/wiki/Nilpotent" title="Nilpotent">nilpotent</a> e to write <i>x</i> + <i>y</i> e for (<i>x,y</i>). The action of e on <i>P</i> reduces the plane to a line: It can be decomposed into the <a href="/wiki/Projection_(mathematics)" title="Projection (mathematics)">projection</a> into the x-coordinate, then quarter-turning the result to the y-axis: e (<i>x</i> + <i>y</i> e) = <i>x</i> e since e<sup>2</sup> = 0. A number <i>x</i> + <i>y</i> e is a <a href="/wiki/Dual_number" title="Dual number">dual number</a>. The dual numbers form a ring, but, since e has no multiplicative inverse, it does not generate a group so the action is not a group action. </p><p>Excluding (0,0) from <i>P</i> makes [<i>x</i> : <i>y</i>] <a href="/wiki/Projective_coordinates" class="mw-redirect" title="Projective coordinates">projective coordinates</a> which describe the real projective line, a one-dimensional space. Since the origin is excluded, at least one of the ratios <i>x</i>/<i>y</i> and <i>y</i>/<i>x</i> exists. Then [<i>x</i> : <i>y</i>] = [<i>x</i>/<i>y</i> : 1] or [<i>x</i> : <i>y</i>] = [1 : <i>y</i>/<i>x</i>]. The projective line <b>P</b><sup>1</sup>(<b>R</b>) is a <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a> covered by two <a href="/wiki/Topological_manifold#Coordinate_charts" title="Topological manifold">coordinate charts</a>, [<i>z</i> : 1] → <i>z</i> or [1 : <i>z</i>] → <i>z</i>, which form an <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a>. For points covered by both charts the <i>transition function</i> is multiplicative inversion on an open neighborhood of the point, which provides a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> as required in a manifold. One application of the real projective line is found in <a href="/wiki/Cayley%E2%80%93Klein_metric" title="Cayley–Klein metric">Cayley–Klein metric</a> geometry. </p> <div class="mw-heading mw-heading3"><h3 id="n_=_3"><span id="n_.3D_3"></span><i>n</i> = 3</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=18" title="Edit section: n = 3"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Duality_Hexa-Okta_SVG.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Duality_Hexa-Okta_SVG.svg/220px-Duality_Hexa-Okta_SVG.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Duality_Hexa-Okta_SVG.svg/330px-Duality_Hexa-Okta_SVG.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Duality_Hexa-Okta_SVG.svg/440px-Duality_Hexa-Okta_SVG.svg.png 2x" data-file-width="744" data-file-height="744" /></a><figcaption><a href="/wiki/Cube" title="Cube">Cube</a> (the hypercube) and <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> (the cross-polytope) of <span class="texhtml"><b>R</b><sup>3</sup></span>. Coordinates are not shown</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional space</a></div> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="n_=_4"><span id="n_.3D_4"></span><i>n</i> = 4</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=19" title="Edit section: n = 4"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:4-cube_3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/4-cube_3D.png/220px-4-cube_3D.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/4-cube_3D.png/330px-4-cube_3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/4-cube_3D.png/440px-4-cube_3D.png 2x" data-file-width="2100" data-file-height="2100" /></a><figcaption></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four-dimensional space</a></div> <p><span class="texhtml"><b>R</b><sup>4</sup></span> can be imagined using the fact that <a href="/wiki/16_(number)" title="16 (number)">16</a> points <span class="texhtml">(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>)</span>, where each <span class="texhtml mvar" style="font-style:italic;">x<sub>k</sub></span> is either 0 or 1, are vertices of a <a href="/wiki/Tesseract" title="Tesseract">tesseract</a> (pictured), the 4-hypercube (see <a href="#Polytopes_in_Rn">above</a>). </p><p>The first major use of <span class="texhtml"><b>R</b><sup>4</sup></span> is a <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> model: three spatial coordinates plus one <a href="/wiki/Time" title="Time">temporal</a>. This is usually associated with <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a>, although four dimensions were used for such models since <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galilei</a>. The choice of theory leads to different structure, though: in <a href="/wiki/Galilean_relativity" class="mw-redirect" title="Galilean relativity">Galilean relativity</a> the <span class="texhtml mvar" style="font-style:italic;">t</span> coordinate is privileged, but in Einsteinian relativity it is not. Special relativity is set in <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>. General relativity uses curved spaces, which may be thought of as <span class="texhtml"><b>R</b><sup>4</sup></span> with a <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">curved metric</a> for most practical purposes. None of these structures provide a (positive-definite) <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> on <span class="texhtml"><b>R</b><sup>4</sup></span>. </p><p>Euclidean <span class="texhtml"><b>R</b><sup>4</sup></span> also attracts the attention of mathematicians, for example due to its relation to <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>, a 4-dimensional <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">real algebra</a> themselves. See <a href="/wiki/Rotations_in_4-dimensional_Euclidean_space" title="Rotations in 4-dimensional Euclidean space">rotations in 4-dimensional Euclidean space</a> for some information. </p><p>In differential geometry, <span class="texhtml"><i>n</i> = 4</span> is the only case where <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> admits a non-standard <a href="/wiki/Differential_structure" title="Differential structure">differential structure</a>: see <a href="/wiki/Exotic_R4" title="Exotic R4">exotic R<sup>4</sup></a>. </p> <div class="mw-heading mw-heading2"><h2 id="Norms_on_Rn">Norms on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=20" title="Edit section: Norms on Rn"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One could define many norms on the <a href="/wiki/Vector_space" title="Vector space">vector space</a> <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. Some common examples are </p> <ul><li>the <a href="/wiki/P-norm" class="mw-redirect" title="P-norm">p-norm</a>, defined by <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="{\textstyle \|\mathbf {x} \|_{p}:={\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \|\mathbf {x} \|_{p}:={\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf35f8f8bd1290edf59ab62204c0dffefa08e3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.087ex; height:4.843ex;" alt="{\textstyle \|\mathbf {x} \|_{p}:={\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}"></span> for all <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} \in \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22fb63c1e813e5c0fe2f7961ae75c9d683f66fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.473ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}"></span> where <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is a positive integer. The case <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 p=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}"></span> is very important, because it is exactly the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>.</li> <li>the <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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>-norm or <a href="/wiki/Maximum_norm" class="mw-redirect" title="Maximum norm">maximum norm</a>, defined by <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} \|_{\infty }:=\max\{x_{1},\dots ,x_{n}\}}"> <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> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>:=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|_{\infty }:=\max\{x_{1},\dots ,x_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d7b4aa923d84f543e1651c731214c8ad6b9bb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.118ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {x} \|_{\infty }:=\max\{x_{1},\dots ,x_{n}\}}"></span> for all <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} \in \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22fb63c1e813e5c0fe2f7961ae75c9d683f66fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.473ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}"></span>. This is the limit of all the <a href="/wiki/P-norm" class="mw-redirect" title="P-norm">p-norms</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="{\textstyle \|\mathbf {x} \|_{\infty }=\lim _{p\to \infty }{\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \|\mathbf {x} \|_{\infty }=\lim _{p\to \infty }{\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372b0f141ab63d3195ba2eb91c83f792729b74ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.219ex; height:4.843ex;" alt="{\textstyle \|\mathbf {x} \|_{\infty }=\lim _{p\to \infty }{\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}"></span>.</li></ul> <p>A really surprising and helpful result is that every norm defined on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is <a href="/wiki/Equivalent_norm" class="mw-redirect" title="Equivalent norm">equivalent</a>. This means for two arbitrary norms <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 \|\cdot \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"></span> and <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 \|\cdot \|'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f887b8185f7527e0f4afeb93889c268ba7d504c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|'}"></span> on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> you can always find positive real numbers <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 \alpha ,\beta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f650e33744628e414c5e7bc24ecf964e3a39554f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.114ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta >0}"></span>, such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \cdot \|\mathbf {x} \|\leq \|\mathbf {x} \|'\leq \beta \cdot \|\mathbf {x} \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <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">x</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mo>′</mo> </msup> <mo>≤</mo> <mi>β</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \cdot \|\mathbf {x} \|\leq \|\mathbf {x} \|'\leq \beta \cdot \|\mathbf {x} \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e717c1460a68394f829a3c097870cf83b1d72a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.267ex; height:3.009ex;" alt="{\displaystyle \alpha \cdot \|\mathbf {x} \|\leq \|\mathbf {x} \|'\leq \beta \cdot \|\mathbf {x} \|}"></span> for all <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} \in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f33c7feddfbe1e5c87647a55639651d9fb2f23de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.148ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}"></span>. </p><p>This defines an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on the set of all norms on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. With this result you can check that a sequence of vectors in <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> converges with <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 \|\cdot \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"></span> if and only if it converges with <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 \|\cdot \|'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f887b8185f7527e0f4afeb93889c268ba7d504c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|'}"></span>. </p><p>Here is a sketch of what a proof of this result may look like: </p><p>Because of the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> it is enough to show that every norm on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> is equivalent to the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</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 \|\cdot \|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a8e44a2eb980f856968a6357e3d0a7c22c905f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.058ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{2}}"></span>. Let <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 \|\cdot \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"></span> be an arbitrary norm on <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. The proof is divided in two steps: </p> <ul><li>We show that there exists 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 \beta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a87dc52878418173659e6d0ff8e77ab2897eac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >0}"></span>, such that <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} \|\leq \beta \cdot \|\mathbf {x} \|_{2}}"> <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 fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mi>β</mi> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|\leq \beta \cdot \|\mathbf {x} \|_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d583ba112b4af6c10934bc7d6073501c1d12969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.636ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {x} \|\leq \beta \cdot \|\mathbf {x} \|_{2}}"></span> for all <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} \in \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22fb63c1e813e5c0fe2f7961ae75c9d683f66fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.473ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}"></span>. In this step you use the fact that every <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} =(x_{1},\dots ,x_{n})\in \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})\in \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9591e345c6ee43e68571426e6db8c66f34aaee23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.491ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})\in \mathbf {R} ^{n}}"></span> can be represented as a linear combination of the standard <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</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="{\textstyle \mathbf {x} =\sum _{i=1}^{n}e_{i}\cdot x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>e</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {x} =\sum _{i=1}^{n}e_{i}\cdot x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44bfe1f99ea6b71b129ac73bf36071b9afcde659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.942ex; height:3.176ex;" alt="{\textstyle \mathbf {x} =\sum _{i=1}^{n}e_{i}\cdot x_{i}}"></span>. Then with the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} \|=\left\|\sum _{i=1}^{n}e_{i}\cdot x_{i}\right\|\leq \sum _{i=1}^{n}\|e_{i}\|\cdot |x_{i}|\leq {\sqrt {\sum _{i=1}^{n}\|e_{i}\|^{2}}}\cdot {\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}=\beta \cdot \|\mathbf {x} \|_{2},}"> <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 fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>e</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> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>≤</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi>β</mi> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|=\left\|\sum _{i=1}^{n}e_{i}\cdot x_{i}\right\|\leq \sum _{i=1}^{n}\|e_{i}\|\cdot |x_{i}|\leq {\sqrt {\sum _{i=1}^{n}\|e_{i}\|^{2}}}\cdot {\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}=\beta \cdot \|\mathbf {x} \|_{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/899b43029143202dfffb87db2aeb3e7c37fc99cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.939ex; height:7.509ex;" alt="{\displaystyle \|\mathbf {x} \|=\left\|\sum _{i=1}^{n}e_{i}\cdot x_{i}\right\|\leq \sum _{i=1}^{n}\|e_{i}\|\cdot |x_{i}|\leq {\sqrt {\sum _{i=1}^{n}\|e_{i}\|^{2}}}\cdot {\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}=\beta \cdot \|\mathbf {x} \|_{2},}"></span> where <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="{\textstyle \beta :={\sqrt {\sum _{i=1}^{n}\|e_{i}\|^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>β</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \beta :={\sqrt {\sum _{i=1}^{n}\|e_{i}\|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de8335822a57c325f2cf84d02ebe164fb33d66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.404ex; height:4.843ex;" alt="{\textstyle \beta :={\sqrt {\sum _{i=1}^{n}\|e_{i}\|^{2}}}}"></span>.</li> <li>Now we have to find an <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 \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span>, such that <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 \alpha \cdot \|\mathbf {x} \|_{2}\leq \|\mathbf {x} \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \cdot \|\mathbf {x} \|_{2}\leq \|\mathbf {x} \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b2ce27e397553d03bb66978b2f408e361bb52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.791ex; height:2.843ex;" alt="{\displaystyle \alpha \cdot \|\mathbf {x} \|_{2}\leq \|\mathbf {x} \|}"></span> for all <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} \in \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22fb63c1e813e5c0fe2f7961ae75c9d683f66fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.473ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}"></span>. Assume there is no such <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. Then there exists for every <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 k\in \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbf {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be3860fd88d718416a1674a9efc68b18a80f06ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.143ex; height:2.176ex;" alt="{\displaystyle k\in \mathbf {N} }"></span> 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 \mathbf {x} _{k}\in \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{k}\in \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126abd9499db721a6ee62191a8bdf4d712f26178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.562ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} _{k}\in \mathbf {R} ^{n}}"></span>, such that <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} _{k}\|_{2}>k\cdot \|\mathbf {x} _{k}\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mi>k</mi> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} _{k}\|_{2}>k\cdot \|\mathbf {x} _{k}\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66a67ae2b8e0eacb10a3bb9c0618645f267cfa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.692ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {x} _{k}\|_{2}>k\cdot \|\mathbf {x} _{k}\|}"></span>. Define a second sequence <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 ({\tilde {\mathbf {x} }}_{k})_{k\in \mathbf {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\tilde {\mathbf {x} }}_{k})_{k\in \mathbf {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0d22b0909dc6d31244b95321ce1397fb7711f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.973ex; height:2.843ex;" alt="{\displaystyle ({\tilde {\mathbf {x} }}_{k})_{k\in \mathbf {N} }}"></span> by <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="{\textstyle {\tilde {\mathbf {x} }}_{k}:={\frac {\mathbf {x} _{k}}{\|\mathbf {x} _{k}\|_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tilde {\mathbf {x} }}_{k}:={\frac {\mathbf {x} _{k}}{\|\mathbf {x} _{k}\|_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d17f313e1ed82680a5b53233f115cd13965cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.414ex; height:4.176ex;" alt="{\textstyle {\tilde {\mathbf {x} }}_{k}:={\frac {\mathbf {x} _{k}}{\|\mathbf {x} _{k}\|_{2}}}}"></span>. This sequence is bounded because <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 \|{\tilde {\mathbf {x} }}_{k}\|_{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\tilde {\mathbf {x} }}_{k}\|_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3103962182b88605c8c2e9de38aaf04b795563ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.14ex; height:2.843ex;" alt="{\displaystyle \|{\tilde {\mathbf {x} }}_{k}\|_{2}=1}"></span>. So because of the <a href="/wiki/Bolzano%E2%80%93Weierstrass_theorem" title="Bolzano–Weierstrass theorem">Bolzano–Weierstrass theorem</a> there exists a convergent subsequence <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 ({\tilde {\mathbf {x} }}_{k_{j}})_{j\in \mathbf {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\tilde {\mathbf {x} }}_{k_{j}})_{j\in \mathbf {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ed765a614001ff6be73f7d09f10b537325e703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.508ex; height:3.176ex;" alt="{\displaystyle ({\tilde {\mathbf {x} }}_{k_{j}})_{j\in \mathbf {N} }}"></span> with limit <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 {a} \in }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>∈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \in }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8af35f5c414aeb06e9c4b19a3523ff4f37dc3bec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.495ex; height:1.843ex;" alt="{\displaystyle \mathbf {a} \in }"></span> <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. Now we show that <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 {a} \|_{2}=1}"> <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">a</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} \|_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6e7c2e745a82ac1e6edbe9ad2cc3de8e49f729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.94ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {a} \|_{2}=1}"></span> but <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 {a} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966e2b83e48cb9c815b1e13aeda3a64141fa38b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.735ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} =\mathbf {0} }"></span>, which is a contradiction. It is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {a} \|\leq \left\|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\right\|+\left\|{\tilde {\mathbf {x} }}_{k_{j}}\right\|\leq \beta \cdot \left\|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}+{\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}\ {\overset {j\to \infty }{\longrightarrow }}\ 0,}"> <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">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>+</mo> <mrow> <mo symmetric="true">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mo>≤</mo> <mi>β</mi> <mo>⋅</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">⟶</mo> <mrow> <mi>j</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </mover> </mrow> <mtext> </mtext> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} \|\leq \left\|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\right\|+\left\|{\tilde {\mathbf {x} }}_{k_{j}}\right\|\leq \beta \cdot \left\|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}+{\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}\ {\overset {j\to \infty }{\longrightarrow }}\ 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9608dfc60c23ac9364ab658bf67248d8fde63b6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.827ex; height:7.176ex;" alt="{\displaystyle \|\mathbf {a} \|\leq \left\|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\right\|+\left\|{\tilde {\mathbf {x} }}_{k_{j}}\right\|\leq \beta \cdot \left\|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}+{\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}\ {\overset {j\to \infty }{\longrightarrow }}\ 0,}"></span> because <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 {a} -{\tilde {\mathbf {x} }}_{k_{j}}\|\to 0}"> <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">a</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\|\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40acdd2a985188dd7b3ac914c347a637263a915b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.455ex; height:3.176ex;" alt="{\displaystyle \|\mathbf {a} -{\tilde {\mathbf {x} }}_{k_{j}}\|\to 0}"></span> and <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 0\leq {\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}<{\frac {1}{k_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq {\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}<{\frac {1}{k_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b994f69e0a8121ceb7974a9566b6a9019722d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.746ex; height:7.176ex;" alt="{\displaystyle 0\leq {\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}<{\frac {1}{k_{j}}}}"></span>, so <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 {\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28bcf624288f8d8484dadc64b601ce316a15527f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.206ex; height:7.176ex;" alt="{\displaystyle {\frac {\|\mathbf {x} _{k_{j}}\|}{\|\mathbf {x} _{k_{j}}\|_{2}}}\to 0}"></span>. This implies <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 {a} \|=0}"> <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">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} \|=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87cbce161ef067c501c2158bb40c04740f091ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.885ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {a} \|=0}"></span>, so <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 {a} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966e2b83e48cb9c815b1e13aeda3a64141fa38b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.735ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} =\mathbf {0} }"></span>. On the other hand <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 {a} \|_{2}=1}"> <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">a</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} \|_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6e7c2e745a82ac1e6edbe9ad2cc3de8e49f729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.94ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {a} \|_{2}=1}"></span>, because <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 {a} \|_{2}=\left\|\lim _{j\to \infty }{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}=\lim _{j\to \infty }\left\|{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}=1}"> <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">a</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mrow> <mo symmetric="true">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} \|_{2}=\left\|\lim _{j\to \infty }{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}=\lim _{j\to \infty }\left\|{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa150a5efb4cc019a434d5af82475a3f181d1d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.024ex; height:5.843ex;" alt="{\displaystyle \|\mathbf {a} \|_{2}=\left\|\lim _{j\to \infty }{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}=\lim _{j\to \infty }\left\|{\tilde {\mathbf {x} }}_{k_{j}}\right\|_{2}=1}"></span>. This can not ever be true, so the assumption was false and there exists such 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 \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Exponential_object" title="Exponential object">Exponential object</a>, for theoretical explanation of the superscript notation</li> <li><a href="/wiki/Geometric_space" class="mw-redirect" title="Geometric space">Geometric space</a></li> <li><a href="/wiki/Real_projective_space" title="Real projective space">Real projective space</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_coordinate_space&action=edit&section=22" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKelley,_John_L.1975" class="citation book cs1">Kelley, John L. (1975). <i>General Topology</i>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90125-6" title="Special:BookSources/0-387-90125-6"><bdi>0-387-90125-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.pub=Springer-Verlag&rft.date=1975&rft.isbn=0-387-90125-6&rft.au=Kelley%2C+John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+coordinate+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkres,_James1999" class="citation book cs1">Munkres, James (1999). <i>Topology</i>. Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-181629-2" title="Special:BookSources/0-13-181629-2"><bdi>0-13-181629-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.pub=Prentice-Hall&rft.date=1999&rft.isbn=0-13-181629-2&rft.au=Munkres%2C+James&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+coordinate+space" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist 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Real coordinate space - Wikipedia