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| In [[mathematical analysis]], the '''Hilbert–Schmidt theorem''', also known as the '''[[eigenfunction]] expansion theorem''', is a fundamental result concerning [[compact operator|compact]], [[self-adjoint operator]]s on [[Hilbert space]]s. In the theory of [[partial differential equation]]s, it is very useful in solving [[elliptic operator|elliptic]] [[boundary value problem]]s.
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| ==Statement of the theorem==
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| Let (''H'', ⟨ , ⟩) be a [[real number|real]] or [[complex number|complex]] Hilbert space and let ''A'' : ''H'' → ''H'' be a [[bounded linear operator|bounded]], compact, self-adjoint operator. Then there is a sequence of non-zero real [[eigenvalue]]s ''λ''<sub>''i''</sub>, ''i'' = 1, ..., ''N'', with ''N'' equal to the [[rank (linear algebra)|rank]] of ''A'', such that |''λ''<sub>''i''</sub>| is [[monotone sequence|monotonically non-increasing]] and, if ''N'' = +∞,
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| :<math>\lim_{i \to + \infty} \lambda_{i} = 0.</math> | |
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| Furthermore, if each eigenvalue of ''A'' is repeated in the sequence according to its [[Multiplicity (mathematics)|multiplicity]], then there exists an [[orthonormal]] set ''φ''<sub>''i''</sub>, ''i'' = 1, ..., ''N'', of corresponding eigenfunctions, i.e.
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| :<math>A \varphi_{i} = \lambda_{i} \varphi_{i} \mbox{ for } i = 1, \dots, N.</math>
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| Moreover, the functions ''φ''<sub>''i''</sub> form an [[orthonormal basis]] for the [[range (mathematics)|range]] of ''A'' and ''A'' can be written as
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| :<math>A u = \sum_{i = 1}^{N} \lambda_{i} \langle \varphi_{i}, u \rangle \varphi_{i} \mbox{ for all } u \in H.</math>
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| ==References==
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| * {{cite book
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| | author = Renardy, Michael and Rogers, Robert C.
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| | title = An introduction to partial differential equations
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| | series = Texts in Applied Mathematics 13
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| | edition = Second edition
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| |publisher = Springer-Verlag
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| | location = New York
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| | year = 2004
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| | pages = 356
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| | isbn = 0-387-00444-0
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| }} (Theorem 8.94)
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| {{DEFAULTSORT:Hilbert-Schmidt theorem}}
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| [[Category:Operator theory]]
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| [[Category:Theorems in functional analysis]]
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23 yr old Boarding Run or Cattery Operator Blomquist from Wallaceburg, likes to spend time skeet shooting, ganhando dinheiro na internet and cigar smoking. In the last few months has made a trip to spots like Bwindi Impenetrable National Park.
Review my homepage: ganhar dinheiro