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| In [[mathematics]], the '''Riesz–Fischer theorem''' in [[real analysis]] is any of a number of closely related results concerning the properties of the space [[Lp space|''L''<sup>2</sup>]] of [[square integrable]] functions. The theorem was proven independently in 1907 by [[Frigyes Riesz]] and [[Ernst Sigismund Fischer]].
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| For many authors, the Riesz–Fischer theorem refers to the fact that the [[Lp space|''L''<sup>''p''</sup> spaces]] from [[Lebesgue integration]] theory are [[Complete metric space|complete]].
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| == Modern forms of the theorem ==
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| The most common form of the theorem states that a measurable function on [–π, π] is [[square integrable]] [[if and only if]] the corresponding [[Fourier series]] converges in the [[Lp space|space ''L''<sup>2</sup>]]. This means that if the ''N''th [[partial sum]] of the Fourier series corresponding to a square-integrable function ''f'' is given by
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| :<math>S_N f(x) = \sum_{n=-N}^{N} F_n \, \mathrm{e}^{inx},</math>
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| where ''F''<sub>''n''</sub>, the ''n''th Fourier [[coefficient]], is given by
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| :<math>F_n =\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\, \mathrm{e}^{-inx}\, \mathrm{d}x,</math>
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| then
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| :<math>\lim_{N \to \infty} \left \Vert S_N f - f \right \|_2 = 0,</math>
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| where <math>\left \Vert \cdot \right \|_2</math> is the ''L''<sup>2</sup>-[[norm (mathematics)|norm]].
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| Conversely, if <math>\left \{ a_n \right \} \,</math> is a two-sided [[sequence]] of [[complex number]]s (that is, its [[Indexed family|indices]] range from negative [[infinity]] to positive infinity) such that
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| :<math>\sum_{n=-\infty}^\infty \left | a_n \right \vert^2 < \infty,</math>
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| then there exists a function ''f'' such that ''f'' is square-integrable and the values <math>a_n</math> are the Fourier coefficients of ''f''.
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| This form of the Riesz–Fischer theorem is a stronger form of [[Bessel's inequality]], and can be used to prove [[Parseval's identity]] for [[Fourier series]].
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| Other results are often called the Riesz–Fischer theorem {{harv|Dunford|Schwartz|1958|loc=§IV.16}}. Among them is the theorem that, if ''A'' is an [[orthonormal]] set in a [[Hilbert space]] ''H'', and ''x'' ∈ ''H'', then
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| :<math>\langle x, y\rangle = 0</math>
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| for all but countably many ''y'' ∈ ''A'', and
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| :<math>\sum_{y\in A} |\langle x,y\rangle|^2 \le \|x\|^2.</math>
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| Furthermore, if ''A'' is an orthonormal basis for ''H'' and ''x'' an arbitrary vector, the series
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| :<math>\sum_{y\in A} \langle x,y\rangle \, y</math>
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| converges ''commutatively'' (or ''unconditionally'') to ''x''. This is equivalent to saying that for every ''ε'' > 0, there exists a finite set ''B''<sub>0</sub> in ''A'' such that
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| :<math> \|x - \sum_{y\in B} \langle x,y\rangle y \| < \varepsilon</math>
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| for every finite set ''B'' containing ''B''<sub>0</sub>. Moreover, the following conditions on the set ''A'' are equivalent:
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| * the set ''A'' is an orthonormal basis of ''H''
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| * for every vector ''x'' ∈ ''H'',
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| ::<math>\|x\|^2 = \sum_{y\in A} |\langle x,y\rangle|^2.</math>
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| Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that ''L''<sup>2</sup> (or more generally ''L''<sup>''p''</sup>, 0 < ''p'' ≤ ∞) is [[complete metric space|complete]].
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| == Example ==
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| The Riesz–Fischer theorem also applies in a more general setting. Let ''R'' be an [[inner product]] space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let {<math>\phi_n</math>} be an orthonormal system in ''R'' (e.g. Fourier basis, Hermite or [[Laguerre polynomials]], etc. – see [[orthogonal polynomials]]), not necessarily complete (in an inner product space, an [[orthonormality|orthonormal set]] is [[complete space|complete]] if no nonzero vector is orthogonal to every vector in the set). The theorem asserts that if the normed space ''R'' is complete (thus ''R'' is a [[Hilbert space]]), then any sequence {<math>c_n</math>} that has finite ℓ<sup>2</sup> norm defines a function ''f'' in the space ''R''.
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| The function ''f'' is defined by
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| <math>f = \lim_{n \to \infty} \sum_{k=0}^n c_k \phi_k </math>, limit in ''R''-norm.
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| Combined with the [[Bessel's inequality]], we know the converse as well: if ''f'' is a function in ''R'', then the Fourier coefficients <math>(f,\phi_n)</math> have finite ℓ<sup>2</sup> [[Norm (mathematics)|norm]].
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| == History: the Note of Riesz and the Note of Fischer (1907) ==
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| In his Note, {{Harvtxt|Riesz|1907|p=616}} states the following result (translated here to modern language at one point: the notation ''L''<sup>2</sup>([''a'', ''b'']) was not used in 1907).
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| :''Let {φ<sub>n </sub>} be an orthonormal system in'' ''L''<sup>2</sup>([''a'', ''b'']) ''and {a<sub>n </sub>} a sequence of reals. The convergence of the series <math> \sum a_n^2 </math> is a necessary and sufficient condition for the existence of a function'' ''f'' ''such that''
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| ::<math> \int_a^b f(x) \varphi_n(x) \, \mathrm{d}x = a_n</math>
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| :''for every'' ''n''.
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| Today, this result of Riesz is a special case of basic facts about series of orthogonal vectors in Hilbert spaces.
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| Riesz's Note appeared in March. In May, {{Harvtxt|Fischer|1907|p=1023}} states explicitly in a theorem (almost with modern words) that a [[Cauchy sequence]] in ''L''<sup>2</sup>([''a'', ''b'']) converges in ''L''<sup>2</sup>-norm to some function ''f''  in ''L''<sup>2</sup>([''a'', ''b'']). In this Note, Cauchy sequences are called "''sequences converging in the mean''" and ''L''<sup>2</sup>([''a'', ''b'']) is denoted by ''Ω''. Also, convergence to a limit in ''L''<sup>2</sup>–norm is called "''convergence in the mean towards a function''". Here is the statement, translated from French:
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| :'''Theorem.''' ''If a sequence of functions belonging to Ω  converges in the mean, there exists in Ω a function f towards which the sequence converges in the mean.''
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| Fischer goes on proving the preceding result of Riesz, as a consequence of the orthogonality of the system, and of the completeness of ''L''<sup>2</sup>.
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| Fischer's proof of completeness is somewhat indirect. It uses the fact that the indefinite integrals of the functions ''g<sub>n</sub>'' in the given Cauchy sequence, namely
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| :<math> G_n(x) = \int_a^x g_n(t) \, \mathrm{d}t,</math>
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| converge uniformly on [''a'', ''b''] to some function ''G'', continuous with bounded variation.
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| The existence of the limit ''g'' ∈ ''L''<sup>2</sup> for the Cauchy sequence is obtained by applying to ''G'' differentiation theorems from Lebesgue's theory. <br />
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| Riesz uses a similar reasoning in his Note, but makes no explicit mention to the completeness of ''L''<sup>2</sup>, although his result may be interpreted this way. He says that integrating term by term a trigonometric series with given square summable coefficients, he gets a series converging uniformly to a continuous function ''F''  with bounded variation. The derivative ''f''  of ''F'', defined almost everywhere, is square summable and has for ''Fourier coefficients'' the given coefficients.
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| == Completeness of ''L''<sup>''p''</sup>, 0 < ''p'' ≤ ∞ ==
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| The proof that ''L<sup>p</sup>'' is [[Complete metric space|complete]] is based on the convergence theorems for the [[Lebesgue integration|Lebesgue integral]].
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| When 1 ≤ ''p'' ≤ ∞, the [[Minkowski inequality]] implies that the [[Lp space|space ''L''<sup>''p''</sup>]] is a normed space. In order to prove that ''L''<sup>''p''</sup> is complete, i.e. that ''L''<sup>''p''</sup> is a [[Banach space]], it is enough (see e.g. [[Banach_space#Definition]]) to prove that every series ∑ ''u''<sub>''n''</sub> of functions in ''L''<sup>''p''</sup>(''μ'') such that
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| : <math> \sum \|u_n\|_p < \infty </math> | |
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| converges in the ''L<sup>p</sup>''-norm to some function ''f'' ∈ ''L<sup>p</sup>''(''μ''). For ''p'' < ∞, the Minkowski inequality and the [[monotone convergence theorem]] imply that
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| :<math> \int \Bigl( \sum_{n=0}^\infty |u_n| \Bigr)^p \, \mathrm{d}\mu \le \Bigl( \sum_{n=0}^{\infty} \|u_n\|_p \Bigr)^p< \infty, \ \ \text{ hence } \ \ f = \sum_{n=0}^\infty u_n</math>
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| is defined ''μ''–almost everywhere and ''f'' ∈ ''L''<sup>''p''</sup>(''μ''). The [[dominated convergence theorem]] is then used to prove that the partial sums of the series converge to ''f'' in the ''L''<sup>''p''</sup>-norm,
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| :<math> \int \left| f - \sum_{k=0}^{n} u_k \right|^p \, \mathrm{d}\mu \le \int \left( \sum_{\ell > n} |u_\ell| \right)^p \, \mathrm{d}\mu \rightarrow 0 \text{ as } n \rightarrow \infty.</math>
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| The case 0 < ''p'' < 1 requires some modifications, due to the fact that the ''p''-norm is no longer subadditive. One starts with the stronger assumption that
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| : <math> \sum \|u_n\|_p^p < \infty</math>
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| and uses repeatedly that
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| : <math>\left|\sum_{k=0}^n u_k \right|^p \le \sum_{k=0}^n |u_k|^p \text{ when } p<1 </math>
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| The case ''p'' = ∞ reduces to a simple question about uniform convergence outside a ''μ''-negligible set.
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| == References ==
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| *{{citation|last=Beals|first=Richard|year=2004|title=Analysis: An Introduction|publication-place=New York|publisher=Cambridge University Press|isbn=0-521-60047-2}}.
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| * {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|title=Linear operators, Part I|publisher=Wiley-Interscience|year=1958}}.
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| *{{citation|last=Fischer|first=Ernst|authorlink=Ernst Sigismund Fischer|title=Sur la convergence en moyenne|journal=Comptes rendus de l'Académie des sciences|volume=144|pages=1022–1024|year=1907}}.
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| *{{citation|last=Riesz|first=Frigyes|authorlink=Frigyes Riesz|title=Sur les systèmes orthogonaux de fonctions|journal=Comptes rendus de l'Académie des sciences|year=1907|volume=144|pages=615–619}}.
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| {{DEFAULTSORT:Riesz-Fischer theorem}}
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| [[Category:Fourier series]]
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| [[Category:Theorems in real analysis]]
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