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<title>Compact embedding</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Compact embedding</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>, the notion of being <b>compactly embedded</b> expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general <a href="Topology" title="Topology">topology</a> and <a href="Functional_analysis" title="Functional analysis">functional analysis</a>.

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<div class="mw-heading mw-heading2"><h2 id="Definition_(topological_spaces)">Definition (topological spaces)</h2></div>

<p>Let (<i>X</i>, <i>T</i>) be a <a href="Topological_space" title="Topological space">topological space</a>, and let <i>V</i> and <i>W</i> be <a href="Subset" title="Subset">subsets</a> of <i>X</i>. We say that <i>V</i> is <b>compactly embedded</b> in <i>W</i>, and write <i>V</i> ⊂⊂ <i>W</i>, if

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<ul><li><i>V</i> ⊆ Cl(<i>V</i>) ⊆ Int(<i>W</i>), where Cl(<i>V</i>) denotes the <a href="Closure_(topology)" title="Closure (topology)">closure</a> of <i>V</i>, and Int(<i>W</i>) denotes the <a href="Interior_(topology)" title="Interior (topology)">interior</a> of <i>W</i>; and</li>

<li>Cl(<i>V</i>) is <a href="Compact_space" title="Compact space">compact</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Definition_(normed_spaces)">Definition (normed spaces)</h2></div>

<p>Let <i>X</i> and <i>Y</i> be two <a href="Normed_vector_space" title="Normed vector space">normed vector spaces</a> with norms ||•||<sub><i>X</i></sub> and ||•||<sub><i>Y</i></sub> respectively, and suppose that <i>X</i> ⊆ <i>Y</i>. We say that <i>X</i> is <b>compactly embedded</b> in <i>Y</i>, and write <i>X</i> ⊂⊂ <i>Y</i> or <i>X</i> ⋐ <i>Y</i>, if

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<ul><li><i>X</i> is <a href="Continuously_embedded" class="mw-redirect" title="Continuously embedded">continuously embedded</a> in <i>Y</i>; i.e., there is a constant <i>C</i> such that ||<i>x</i>||<sub><i>Y</i></sub> ≤ <i>C</i>||<i>x</i>||<sub><i>X</i></sub> for all <i>x</i> in <i>X</i>; and</li>

<li>The embedding of <i>X</i> into <i>Y</i> is a <a href="Compact_operator" title="Compact operator">compact operator</a>: any <a href="Bounded_set" title="Bounded set">bounded set</a> in <i>X</i> is <a href="Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a> in <i>Y</i>, i.e. every <a href="Sequence" title="Sequence">sequence</a> in such a bounded set has a <a href="Subsequence" title="Subsequence">subsequence</a> that is <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy</a> in the norm ||•||<sub><i>Y</i></sub>.</li></ul>

<p>If <i>Y</i> is a <a href="Banach_space" title="Banach space">Banach space</a>, an equivalent definition is that the embedding operator (the identity) <i>i</i> : <i>X</i> → <i>Y</i> is a <a href="Compact_operator" title="Compact operator">compact operator</a>.

</p><p>When applied to functional analysis, this version of compact embedding is usually used with <a href="Banach_spaces" class="mw-redirect" title="Banach spaces">Banach spaces</a> of functions. Several of the <a href="Sobolev_inequality" title="Sobolev inequality">Sobolev embedding theorems</a> are compact embedding theorems. When an embedding is not compact, it may possess a related, but weaker, property of <a href="Cocompact_embedding" title="Cocompact embedding">cocompactness</a>.

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

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</style><cite id="CITEREFAdams1975" class="citation book cs1">Adams, Robert A. (1975). <i>Sobolev Spaces</i>. Boston, MA: <a href="Academic_Press" title="Academic Press">Academic Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-044150-1</bdi>.</cite></li>

<li><cite id="CITEREFEvans1998" class="citation book cs1"><a href="Lawrence_C._Evans" title="Lawrence C. Evans">Evans, Lawrence C.</a> (1998). <i>Partial differential equations</i>. Providence, RI: American Mathematical Society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-0772-2</bdi>.</cite></li>

<li><cite id="CITEREFRenardy,_M.Rogers,_R._C.1992" class="citation book cs1">Renardy, M. & Rogers, R. C. (1992). <i>An Introduction to Partial Differential Equations</i>. Berlin: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-97952-2</bdi>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-05" href="https://en.wikipedia.org/wiki/?title=Compact_embedding&oldid=1294018083">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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