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<title>Contraction mapping</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Contraction mapping</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p> In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>contraction mapping</b>, or <b>contraction</b> or <b>contractor</b>, on a <a href="Metric_space" title="Metric space">metric space</a> (<i>M</i>, <i>d</i>) is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <i>f</i> from <i>M</i> to itself, with the property that there is some <a href="Real_number" title="Real number">real number</a> <span class="mwe-math-element mwe-math-element-inline"><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 k<1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 0\leq k<1}</annotation>

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</math></span><img src="./907473f47f1507f50b023871549050cc2bcc5ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.733ex; height:2.343ex;" alt="{\displaystyle 0\leq k<1}" loading="lazy"></span> such that for all <i>x</i> and <i>y</i> in <i>M</i>,

</p><p><span class="mwe-math-element mwe-math-element-block"><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(f(x),f(y))\leq k\,d(x,y).}">

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<annotation encoding="application/x-tex">{\displaystyle d(f(x),f(y))\leq k\,d(x,y).}</annotation>

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The smallest such value of <i>k</i> is called the <b><a href="Lipschitz_constant" class="mw-redirect" title="Lipschitz constant">Lipschitz constant</a></b> of <i>f</i>. Contractive maps are sometimes called <b>Lipschitzian maps</b>. If the above condition is instead satisfied for

<i>k</i> ≤ 1, then the mapping is said to be a <a href="Non-expansive_map" class="mw-redirect" title="Non-expansive map">non-expansive map</a>.

</p><p>More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (<i>M</i>, <i>d</i>) and (<i>N</i>, <i>d'</i>) are two metric spaces, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\rightarrow N}">

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<annotation encoding="application/x-tex">{\displaystyle f:M\rightarrow N}</annotation>

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</math></span><img src="./c6cd11d4656ac58a9d4c9ef09866d91950f6cc7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.336ex; height:2.509ex;" alt="{\displaystyle f:M\rightarrow N}" loading="lazy"></span> is a contractive mapping if there is a constant <span class="mwe-math-element mwe-math-element-inline"><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 k<1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq k<1}</annotation>

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</math></span><img src="./907473f47f1507f50b023871549050cc2bcc5ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.733ex; height:2.343ex;" alt="{\displaystyle 0\leq k<1}" loading="lazy"></span> such that

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle d'(f(x),f(y))\leq k\,d(x,y)}</annotation>

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for all <i>x</i> and <i>y</i> in <i>M</i>.

</p><p>Every contraction mapping is <a href="Lipschitz_continuous" class="mw-redirect" title="Lipschitz continuous">Lipschitz continuous</a> and hence <a href="Uniformly_continuous" class="mw-redirect" title="Uniformly continuous">uniformly continuous</a> (for a Lipschitz continuous function, the constant <i>k</i> is no longer necessarily less than 1).

</p><p>A contraction mapping has at most one <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a>. Moreover, the <a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a> states that every contraction mapping on a <a href="Empty_set" title="Empty set">non-empty</a> <a href="Complete_metric_space" title="Complete metric space">complete metric space</a> has a unique fixed point, and that for any <i>x</i> in <i>M</i> the <a href="Iterated_function" title="Iterated function">iterated function</a> sequence <i>x</i>, <i>f</i> (<i>x</i>), <i>f</i> (<i>f</i> (<i>x</i>)), <i>f</i> (<i>f</i> (<i>f</i> (<i>x</i>))), ... converges to the fixed point. This concept is very useful for <a href="Iterated_function_systems" class="mw-redirect" title="Iterated function systems">iterated function systems</a> where <a href="Convergence_proof_techniques#contraction_mapping" title="Convergence proof techniques">contraction mappings are often used</a>. Banach's fixed-point theorem is also applied in proving the existence of solutions of <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a>, and is used in one proof of the <a href="Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a>.<sup id="cite_ref-shifrin_1-0" class="reference"><a href="#cite_note-shifrin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Contraction mappings play an important role in <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a> problems.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Firmly_non-expansive_mapping">Firmly non-expansive mapping</h2></div>

<p>A non-expansive mapping with <span class="mwe-math-element mwe-math-element-inline"><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=1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>

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</math></span><img src="./6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy"></span> can be generalized to a <b>firmly non-expansive mapping</b> in a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation>

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</math></span><img src="./19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"></span> if the following holds for all <i>x</i> and <i>y</i> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation>

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</math></span><img src="./19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"></span>:

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \|f(x)-f(y)\|^{2}\leq \,\langle x-y,f(x)-f(y)\rangle ,}</annotation>

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where

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle d(x,y)=\|x-y\|.}</annotation>

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</math></span></span>

This is a special case of <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" 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 }" loading="lazy"></span> averaged nonexpansive operators with <span class="mwe-math-element mwe-math-element-inline"><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 =1/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =1/2}</annotation>

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</math></span><img src="./34c4cf68ca8870e08f68c602153b0e32bde0326a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.073ex; height:2.843ex;" alt="{\displaystyle \alpha =1/2}" loading="lazy"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A firmly non-expansive mapping is always non-expansive, via the <a href="Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>.

</p><p>The class of firmly non-expansive maps is closed under <a href="Convex_combination" title="Convex combination">convex combinations</a>, but not compositions.<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This class includes <a href="Proximal_operator" title="Proximal operator">proximal mappings</a> of proper, convex, lower-semicontinuous functions, hence it also includes orthogonal <a href="Projection_(mathematics)" title="Projection (mathematics)">projections</a> onto non-empty closed <a href="Convex_set" title="Convex set">convex sets</a>. The class of firmly nonexpansive operators is equal to the set of resolvents of maximally <a href="Monotonic_function#Monotonicity_in_functional_analysis" title="Monotonic function">monotone operators</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Surprisingly, while iterating non-expansive maps has no guarantee to find a fixed point (e.g. multiplication by -1), firm non-expansiveness is sufficient to <a href="Convergence_proof_techniques" title="Convergence proof techniques">guarantee global convergence</a> to a fixed point, provided a fixed point exists. More precisely, if

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fix} f:=\{x\in {\mathcal {H}}\ |\ f(x)=x\}\neq \varnothing ,}</annotation>

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then for any initial point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in {\mathcal {H}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in {\mathcal {H}}}</annotation>

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</math></span><img src="./a553eb0a025dc67629d6b5e05be263435f8eafe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.188ex; height:2.509ex;" alt="{\displaystyle x_{0}\in {\mathcal {H}}}" loading="lazy"></span>, iterating

<msub>

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</msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x_{n+1}=f(x_{n}),\quad \forall n\in \mathbb {N} }</annotation>

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</math></span></span>

yields convergence to a fixed point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\to z\in \operatorname {Fix} f}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}\to z\in \operatorname {Fix} f}</annotation>

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</math></span><img src="./8bd95b569bc15ae763f0052b9b85b8158a6f6109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.149ex; height:2.509ex;" alt="{\displaystyle x_{n}\to z\in \operatorname {Fix} f}" loading="lazy"></span>. This convergence might be <a href="Weak_convergence_(Hilbert_space)" title="Weak convergence (Hilbert space)">weak</a> in an infinite-dimensional setting.<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Subcontraction_map">Subcontraction map</h2></div>

<p>A <b>subcontraction map</b> or <b>subcontractor</b> is a map <i>f</i> on a metric space (<i>M</i>, <i>d</i>) such that

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d(f(x),f(y))\leq d(x,y);}</annotation>

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<mtext>unless</mtext>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle d(f(f(x)),f(x))<d(f(x),x)\quad {\text{unless}}\quad x=f(x).}</annotation>

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</math></span></span>

If the <a href="Image_(mathematics)" title="Image (mathematics)">image</a> of a subcontractor <i>f</i> is <a href="Compact_space" title="Compact space">compact</a>, then <i>f</i> has a fixed point.<sup id="cite_ref-Gold17_7-0" class="reference"><a href="#cite_note-Gold17-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Locally_convex_spaces">Locally convex spaces</h2></div>

<p>In a <a href="Locally_convex_space" class="mw-redirect" title="Locally convex space">locally convex space</a> (<i>E</i>, <i>P</i>) with <a href="Topological_space" title="Topological space">topology</a> given by a set <i>P</i> of <a href="Seminorm" title="Seminorm">seminorms</a>, one can define for any <i>p</i> ∈ <i>P</i> a <i>p</i>-contraction as a map <i>f</i> such that there is some <i>k</i><sub><i>p</i></sub> < 1 such that <span class="nowrap"><i>p</i>(<i>f</i>(<i>x</i>) − <i>f</i>(<i>y</i>))</span> ≤ <span class="nowrap"><i>k<sub>p</sub> p</i>(<i>x</i> − <i>y</i>)</span>. If <i>f</i> is a <i>p</i>-contraction for all <i>p</i> ∈ <i>P</i> and (<i>E</i>, <i>P</i>) is sequentially complete, then <i>f</i> has a fixed point, given as limit of any sequence <i>x</i><sub><i>n</i>+1</sub> = <i>f</i>(<i>x</i><sub><i>n</i></sub>), and if (<i>E</i>, <i>P</i>) is <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a>, then the fixed point is unique.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Short_map" class="mw-redirect" title="Short map">Short map</a></li>

<li><a href="Contraction_(operator_theory)" title="Contraction (operator theory)">Contraction (operator theory)</a></li>

<li><a href="Transformation_(function)" title="Transformation (function)">Transformation</a></li>

<li><a href="Comparametric_equation" title="Comparametric equation">Comparametric equation</a></li>

<li><a href="Blackwell's_contraction_mapping_theorem" title="Blackwell's contraction mapping theorem">Blackwell's contraction mapping theorem</a></li>

<li><a href="CLRg_property" title="CLRg property">CLRg property</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFShifrin2005" class="citation book cs1">Shifrin, Theodore (2005). <i>Multivariable Mathematics</i>. Wiley. pp. <span class="nowrap">244–</span>260. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-52638-4</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFDenardo1967" class="citation journal cs1">Denardo, Eric V. (1967). "Contraction Mappings in the Theory Underlying Dynamic Programming". <i>SIAM Review</i>. <b>9</b> (2): <span class="nowrap">165–</span>177. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967SIAMR...9..165D">1967SIAMR...9..165D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1009030">10.1137/1009030</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFStokeyLucas1989" class="citation book cs1"><a href="Nancy_Stokey" title="Nancy Stokey">Stokey, Nancy L.</a>; <a href="Robert_Lucas_Jr." title="Robert Lucas Jr.">Lucas, Robert E.</a> (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BgQ3AwAAQBAJ&pg=PA49"><i>Recursive Methods in Economic Dynamics</i></a>. Cambridge: Harvard University Press. pp. <span class="nowrap">49–</span>55. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-674-75096-8</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFCombettes2004" class="citation journal cs1">Combettes, Patrick L. (2004). "Solving monotone inclusions via compositions of nonexpansive averaged operators". <i>Optimization</i>. <b>53</b> (<span class="nowrap">5–</span>6): <span class="nowrap">475–</span>504. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F02331930412331327157">10.1080/02331930412331327157</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:219698493">219698493</a>.</cite></span>

</li>

<li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBauschke2017" class="citation book cs1">Bauschke, Heinz H. (2017). <i>Convex Analysis and Monotone Operator Theory in Hilbert Spaces</i>. New York: Springer.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFCombettes2018" class="citation journal cs1">Combettes, Patrick L. (July 2018). "Monotone operator theory in convex optimization". <i>Mathematical Programming</i>. <b>B170</b>: <span class="nowrap">177–</span>206. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1802.02694">1802.02694</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018arXiv180202694C">2018arXiv180202694C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10107-018-1303-3">10.1007/s10107-018-1303-3</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:49409638">49409638</a>.</cite></span>

</li>

<li id="cite_note-Gold17-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gold17_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGoldstein1967" class="citation book cs1">Goldstein, A.A. (1967). <i>Constructive real analysis</i>. Harper's Series in Modern Mathematics. New York-Evanston-London: Harper and Row. p. 17. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0189.49703">0189.49703</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFCainNashed1971" class="citation journal cs1">Cain, G. L. Jr.; <a href="Zuhair_Nashed" title="Zuhair Nashed">Nashed, M. Z.</a> (1971). <a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1971.39.581">"Fixed Points and Stability for a Sum of Two Operators in Locally Convex Spaces"</a>. <i>Pacific Journal of Mathematics</i>. <b>39</b> (3): <span class="nowrap">581–</span>592. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1971.39.581">10.2140/pjm.1971.39.581</a></span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFIstratescu1981" class="citation book cs1">Istratescu, Vasile I. (1981). <i>Fixed Point Theory: An Introduction</i>. Holland: D.Reidel. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-90-277-1224-0</bdi>.</cite> provides an undergraduate level introduction.</li>

<li><cite id="CITEREFGranasDugundji2003" class="citation book cs1">Granas, Andrzej; <a href="James_Dugundji" title="James Dugundji">Dugundji, James</a> (2003). <i>Fixed Point Theory</i>. New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-00173-9</bdi>.</cite></li>

<li><cite id="CITEREFKirkSims2001" class="citation book cs1">Kirk, William A.; <a href="Brailey_Sims" title="Brailey Sims">Sims, Brailey</a> (2001). <i>Handbook of Metric Fixed Point Theory</i>. London: Kluwer Academic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7923-7073-4</bdi>.</cite></li>

<li><cite id="CITEREFNaylorSell1982" class="citation book cs1">Naylor, Arch W.; Sell, George R. (1982). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=t3SXs4-KrE0C&pg=PA125"><i>Linear Operator Theory in Engineering and Science</i></a>. Applied Mathematical Sciences. Vol. 40 (Second ed.). New York: Springer. pp. <span class="nowrap">125–</span>134. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90748-2</bdi>.</cite></li></ul>

<ul><li><cite id="CITEREFBullo2022" class="citation book cs1">Bullo, Francesco (2022). <i>Contraction Theory for Dynamical Systems</i>. Kindle Direct Publishing. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>979-8-8366-4680-6</bdi>.</cite></li></ul>

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</style><div id="Metric_spaces_(Category)86" style="font-size:114%;margin:0 4em"><a href="Metric_space" title="Metric space">Metric spaces</a> (Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Metric_space" title="Metric space">Metric space</a></li>

<li><a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li>

<li><a href="Complete_metric_space" title="Complete metric space">Completeness</a></li>

<li><a href="Equivalence_of_metrics" title="Equivalence of metrics">Equivalent metrics</a></li>

<li><a href="Metrizable_space" title="Metrizable space">Metrizable space</a></li>

<li><a href="Triangle_inequality" title="Triangle inequality">Triangle inequality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Baire_category_theorem" title="Baire category theorem">Baire category theorem</a></li>

<li><a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point</a></li>

<li><a href="Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a></li>

<li><a href="Lebesgue's_number_lemma" title="Lebesgue's number lemma">Lebesgue's number lemma</a></li>

<li><a href="Metrization_theorem" class="mw-redirect" title="Metrization theorem">Metrization theorems</a>:

<li><a href="Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov</a></li>

<li><a href="Urysohn's_metrization_theorem" class="mw-redirect" title="Urysohn's metrization theorem">Urysohn's</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Metric_map" title="Metric map">Metric map</a></li></ul></li>

<li><a href="Dilation_(metric_space)" title="Dilation (metric space)">Dilation</a></li>

<li><a href="Equicontinuity" title="Equicontinuity">Equicontinuity</a></li>

<li>(<a href="Quasi-isometry" title="Quasi-isometry">Quasi-</a>) <a href="Isometry" title="Isometry">Isometry</a></li>

<li><a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a></li>

<li><a href="Metric_derivative" title="Metric derivative">Metric derivative</a></li>

<li><a href="Metric_outer_measure" title="Metric outer measure">Metric outer measure</a></li>

<li><a href="Metric_projection" title="Metric projection">Metric projection</a></li>

<li><a href="Motion_(geometry)" title="Motion (geometry)">Motion</a></li>

<li><a href="Quasisymmetric_map" title="Quasisymmetric map">Quasisymmetric</a></li>

<li><a href="Stretch_factor" title="Stretch factor">Stretch factor</a></li>

<li><a href="Uniform_continuity" title="Uniform continuity">Uniform continuity</a>

<ul><li><a href="Uniform_isomorphism" title="Uniform isomorphism">Isomorphism</a></li></ul></li>

<li><a href="Uniform_convergence" title="Uniform convergence">Uniform convergence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br>metric spaces</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Complete_metric_space" title="Complete metric space">Complete</a></li>

<li><a href="Convex_metric_space" title="Convex metric space">Convex</a></li>

<li><a href="Doubling_space" title="Doubling space">Doubling</a></li>

<li><a href="Hyperbolic_metric_space" title="Hyperbolic metric space">Hyperbolic</a></li>

<li><a href="Injective_metric_space" title="Injective metric space">Injective</a></li>

<li><a href="Length_metric_space" class="mw-redirect" title="Length metric space">Length metric space</a></li>

<li><a href="Metric_space_aimed_at_its_subspace" title="Metric space aimed at its subspace">Metric space aimed at its subspace</a></li>

<li><a href="Polish_space" title="Polish space">Polish</a></li>

<li><a href="Totally_bounded_space" title="Totally bounded space">Totally bounded</a></li>

<li><a href="Tree-graded_space" title="Tree-graded space">Tree-graded</a></li>

<li><a href="Ultrametric_space" title="Ultrametric space">Ultrametric space</a></li>

<li><a href="Uniformly_disconnected_space" title="Uniformly disconnected space">Uniformly disconnected</a></li>

<li><a href="Urysohn_universal_space" title="Urysohn universal space">Urysohn universal</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ball_(mathematics)" title="Ball (mathematics)">Balls</a></li>

<li><a href="Borel_set" title="Borel set">Borel</a></li>

<li><a href="Bounded_set" title="Bounded set">Bounded</a></li>

<li><a href="Delone_set" title="Delone set">Delone</a></li>

<li><a href="Diameter_of_a_set" title="Diameter of a set">Diameter</a></li>

<li><a href="Distance_set" title="Distance set">Distance set</a></li>

<li><a href="Gromov_product" title="Gromov product">Gromov product</a></li>

<li><a href="Gromov%E2%80%93Hausdorff_convergence" title="Gromov–Hausdorff convergence">Gromov–Hausdorff convergence</a></li>

<li><a href="Hausdorff_distance" title="Hausdorff distance">Hausdorff distance</a></li>

<li><a href="Kuratowski_convergence" title="Kuratowski convergence">Kuratowski convergence</a></li>

<li><a href="Meyer_set" title="Meyer set">Meyer</a></li>

<li><a href="Packing_dimension" title="Packing dimension">Packing dimension</a></li>

<li><a href="Porous_set" title="Porous set">Porous</a></li>

<li><a href="Positively_separated_sets" title="Positively separated sets">Positively separated sets</a></li>

<li><a href="Tight_span" title="Tight span">Tight span</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Manifold" title="Manifold">Manifolds</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li>

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a><br>and <a href="Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li>

<li><a href="Inner_product_space" title="Inner product space">Inner product space</a></li>

<li><a href="L%C3%A9vy_metric" title="Lévy metric">Lévy metric</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable topological vector space</a></li>

<li><a href="Normed_space" class="mw-redirect" title="Normed space">Normed space</a></li>

<li><a href="Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li>

<li><a href="Wasserstein_metric" title="Wasserstein metric">Wasserstein metric</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="General_topology" title="General topology">General topology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Discrete_space" title="Discrete space">Discrete space</a></li>

<li><a href="Intrinsic_metric" title="Intrinsic metric">Intrinsic metric</a></li>

<li><a href="Laakso_space" title="Laakso space">Laakso space</a></li>

<li><a href="Product_metric" title="Product metric">Product metric</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Category_of_metric_spaces" title="Category of metric spaces">Category of metric spaces</a></li>

<li><a href="Cantor_space" title="Cantor space">Cantor space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approach_space" title="Approach space">Approach space</a></li>

<li><a href="Cauchy_space" title="Cauchy space">Cauchy space</a></li>

<li><a href="Coarse_structure" title="Coarse structure">Coarse structure</a></li>

<li><a href="Cosmic_space" title="Cosmic space">Cosmic space</a></li>

<li><a href="Diversity_(mathematics)" title="Diversity (mathematics)">Diversity</a></li>

<li><a href="Generalised_metric" title="Generalised metric">Generalised metric</a></li>

<li><a href="Measure_space" title="Measure space">Measure space</a></li>

<li><a href="Probabilistic_metric_space" title="Probabilistic metric space">Probabilistic metric space</a></li>

<li><a href="Proximity_space" title="Proximity space">Proximity space</a></li>

<li><a href="Pseudometric_space" title="Pseudometric space">Pseudometric space</a></li>

<li><a href="Uniform_space" title="Uniform space">Uniform space</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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