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<li><strong><a class="existingWikiWord" href="/nlab/show/functional+analysis">Functional Analysis</a></strong></li> </ul> <h2 id="overview_diagrams">Overview diagrams</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TVS+relationships">topological vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diagram+of+LCTVS+properties">locally convex topological vector spaces</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+convex+topological+vector+space">locally convex topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Banach+space">Banach Spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflexive+Banach+space">reflexive</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Smith+space+%28functional+analysis%29">Smith Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Sobolev+space">Sobolev spaces</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bornological+vector+space">Bornological Vector Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/barrelled+topological+vector+space">Barrelled Vector Spaces</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded</a>, <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded</a>, <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum of an operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-Weierstrass+theorem">Stone-Weierstrass theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></p> </li> </ul> <h2 id="topics_in_functional_analysis">Topics in Functional Analysis</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">Bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theories+in+functional+analysis">Algebraic Theories in Functional Analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">An Elementary Treatment of Hilbert Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+classes+of+Banach+spaces">When are two Banach spaces isomorphic?</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/functional+analysis+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#versions_of_the_theorem'>Versions of the theorem</a></li> <ul> <li><a href='#memory_hook'>Memory hook</a></li> <li><a href='#selfadjoint_operators'>Selfadjoint operators</a></li> <li><a href='#borel_functional_calculus'>Borel functional calculus</a></li> <li><a href='#compact_operators'>Compact operators</a></li> <li><a href='#normal_operators'>Normal operators</a></li> </ul> <li><a href='#proofs'>Proofs</a></li> <ul> <li><a href='#'>…</a></li> <li><a href='#by_lagrange_multipliers'>By Lagrange multipliers</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <strong>spectral theorems</strong> form a cornerstone of <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a>. They are a vast generalization to infinite-dimensional <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> of a basic result in linear algebra: an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> Hermitian matrix can be diagonalized or conjugated to a diagonal matrix with real entries along the diagonal.</p> <p>There is a caveat, though: if we consider a separable Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> then we can choose a countable orthonormal (Hilbert) basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>e</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e_n\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>, a linear operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> then has a matrix representation in this basis just as in finite dimensional linear algebra. The spectral theorem does <em>not</em> say that for every selfadjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> there is a basis so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has a diagonal matrix with respect to it. The situation that comes closest to the analogy of finite dimensions is the spectral theorem for compact operators.</p> <p>There are several versions of the spectral theorem, or several spectral theorems, differing in the kind of operator considered (bounded or unbounded, selfadjoint or normal) and the phrasing of the statement (via spectral measures, multiplication operator norm), which is why this page does not consist of <em>one</em> statement only. Plus there are several different strategies to prove e.g. the spectral theorem for bounded selfadjoint operators, so there is no <em>canonical</em> way to prove it (none is particularly simple).</p> <h2 id="versions_of_the_theorem">Versions of the theorem</h2> <h3 id="memory_hook">Memory hook</h3> <p>This is a short description of a way to prove the spectral theorem for bounded, selfadjoint operators in the spectral measure form, it may serve as a memory hook:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> be a Hilbert space and A be an selfadjoint operator on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>. Then we can form the smallest <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> generated by both A and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\mathbb{1}</annotation></semantics></math>, the identity operator. This is an abelian algebra, that is, via the <span class="newWikiWord">Gelfand representation<a href="/nlab/new/Gelfand+representation">?</a></span>, the abelian version of the <a class="existingWikiWord" href="/nlab/show/Gelfand-Naimark+theorem">Gelfand-Naimark theorem</a>, isometric isomorph to the algebra of continuous complex valued functions of a compact Hausdorff space. The spectral integral of A now becomes the Lebesgue (or Riemann) integral of a continuous function.</p> <h3 id="selfadjoint_operators">Selfadjoint operators</h3> <ul> <li>theorem: there is a one to one correspondence of (bounded or unbounded) selfadjoint operators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/spectral+measure">spectral measure</a>s E such that<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo>∫</mo><mi>λ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>d</mi><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A = \integral \lambda E(d\lambda) </annotation></semantics></math></div> <p>A is bounded iff E is bounded.</p> </li> </ul> <h3 id="borel_functional_calculus">Borel functional calculus</h3> <p>As stated on the <a class="existingWikiWord" href="/nlab/show/spectral+measure">spectral measure</a> page, given a resolution of the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> one can make sense of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(u)</annotation></semantics></math> for any function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> that is Borel measurable. Since we have for a selfadjoint operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = E(u)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">u(\lambda) = \lambda</annotation></semantics></math>, one can define a bounded operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(A)</annotation></semantics></math> for every <em>bounded</em> Borel measurable function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>.</p> <h3 id="compact_operators">Compact operators</h3> <p>The special case of compact operators is mentioned here because it comes closest to the finite dimensional situation of diagonalizing Hermitian matrixes.</p> <ul> <li>theorem: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be a compact selfadjoint operator, then the spectrum of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> consists of eigenvalues only, and all eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\neq 0</annotation></semantics></math> have finite multiplicity. Therefore, the spectral theorem says in this case, that<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><msub><mi>λ</mi> <mi>k</mi></msub><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex"> A = \sum \lambda_k |x\rangle \langle x| </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_k</annotation></semantics></math> is an eigenvalue (each eigenvalue appears n times if n is its multiplicity) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> an eigenvector with eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_k</annotation></semantics></math>.</p> </li> </ul> <p>The converse statement is of course true, too, given a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>λ</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\lambda_n)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> convergent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, the sum above (with arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℋ</mi><mo>,</mo><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x \in \mathcal{H}, \|x\| = 1</annotation></semantics></math>) defines a normal compact operator, and a selfadjoint compact operator iff all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_n</annotation></semantics></math> are real. It is possible to generalize the statement to compact operators on (not necessarily complete) normed spaces.</p> <h3 id="normal_operators">Normal operators</h3> <p>(…) <a class="existingWikiWord" href="/nlab/show/normal+operator">normal operator</a> (…)</p> <h2 id="proofs">Proofs</h2> <h3 id="">…</h3> <h3 id="by_lagrange_multipliers">By Lagrange multipliers</h3> <p>For the case of symmetric real matrices there is a simple proof using <a class="existingWikiWord" href="/nlab/show/Lagrange+multiplier">Lagrange multiplier</a> techniques.</p> <p>See <a href="http://ncatlab.org/nlab/show/Lagrange+multiplier#ApplicationToSpectralTheory">Lagrange multiplier – Applications – To Spectral theory</a>.</p> <h2 id="related_entries">Related entries</h2> <ul> <li>For <a class="existingWikiWord" href="/nlab/show/unitary+matrices">unitary matrices</a> the spectal theorem is equivalently <a href="maximal+torus#Properties">Cartan’s theorem</a> that every element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math> is conjugate to an element in its <a class="existingWikiWord" href="/nlab/show/maximal+torus">maximal torus</a>.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 29, 2014 at 07:29:43. 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