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In [[Mathematics|mathematics]], the '''Riesz–Fischer theorem''' in [[Real analysis|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 [[Biography:Frigyes Riesz|Frigyes Riesz]] and Ernst Sigismund Fischer.

For many authors, the Riesz–Fischer theorem refers to the fact that the [[Lp space]]s <math>L^p</math> from [[Lebesgue integration]] theory are [[Complete metric space|complete]].

== Modern forms of the theorem ==
The most common form of the theorem states that a measurable function on <math>[-\pi, \pi]</math> is square integrable [[If and only if|if and only if]] the corresponding [[Fourier series]] converges in the [[Lp space]] <math>L^2.</math> This means that if the ''N''th partial sum of the Fourier series corresponding to a square-integrable function ''f'' is given by
<math display="block">S_N f(x) = \sum_{n=-N}^{N} F_n \, \mathrm{e}^{inx},</math>
where <math>F_n,</math> the ''n''th Fourier [[Coefficient|coefficient]], is given by
<math display="block">F_n =\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\, \mathrm{e}^{-inx}\, \mathrm{d}x,</math>
then
<math display="block">\lim_{N \to \infty} \left\Vert S_N f - f \right\|_2 = 0,</math>
where <math>\|\,\cdot\,\|_2</math> is the <math>L^2</math>-[[Norm (mathematics)|norm]].

Conversely, if <math> \{a_n\} \,</math> is a two-sided [[Sequence|sequence]] of [[Complex number|complex number]]s (that is, its [[Indexed family|indices]] range from negative [[Infinity|infinity]] to positive infinity) such that
<math display="block">\sum_{n=-\infty}^\infty \left|a_n\right\vert^2 < \infty,</math>
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''.

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]].

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 <math>x \in H,</math> then
<math display="block">\langle x, y\rangle = 0</math>
for all but countably many <math>y \in A,</math> and
<math display="block">\sum_{y\in A} |\langle x,y\rangle|^2 \le \|x\|^2.</math> 
Furthermore, if ''A'' is an orthonormal basis for ''H'' and ''x'' an arbitrary vector, the series
<math display="block">\sum_{y\in A} \langle x,y\rangle \, y</math>
converges {{em|commutatively}} (or {{em|unconditionally}}) to ''x''.  This is equivalent to saying that for every <math>\varepsilon > 0,</math> there exists a finite set <math>B_0</math> in ''A'' such that
<math display="block">\|x - \sum_{y\in B} \langle x,y\rangle y \| < \varepsilon</math>
for every finite set ''B'' containing ''B''<sub>0</sub>. Moreover, the following conditions on the set ''A'' are equivalent:
* the set ''A'' is an orthonormal basis of ''H''
* for every vector <math>x \in H,</math> 
:<math display="block">\|x\|^2 = \sum_{y\in A} |\langle x,y\rangle|^2.</math>

Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that <math>L^2</math> (or more generally <math>L^p, 0 < p \leq \infty</math>) is [[Complete metric space|complete]].

== Example ==
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>\{\varphi_n\}</math> be an orthonormal system in ''R'' (e.g. Fourier basis, Hermite or [[Laguerre polynomials]], etc. – see [[Orthogonal polynomials|orthogonal polynomials]]), not necessarily complete (in an inner product space, an [[Orthonormality|orthonormal set]] is 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 <math>\ell^2</math> norm defines a function ''f'' in the space ''R''.

The function ''f'' is defined by
<math>f = \lim_{n \to \infty} \sum_{k=0}^n c_k \varphi_k,</math> limit in ''R''-norm.

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,\varphi_n)</math> have finite <math>\ell^2</math> [[Norm (mathematics)|norm]].

== History: the Note of Riesz and the Note of Fischer (1907) ==

In his Note, {{Harvtxt|Riesz|1907|p=616}} states the following result (translated here to modern language at one point: the notation <math>L^2([a, b])</math> was not used in 1907).

:''Let'' <math>\left\{\varphi_n\right\}</math>''be an orthonormal system in'' <math>L^2([a, b])</math> ''and'' <math>\left\{a_n\right\}</math> ''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'' <math display="block">\int_a^b f(x) \varphi_n(x) \, \mathrm{d}x = a_n \quad \text{ for every } n.</math>
Today, this result of Riesz is a special case of basic facts about series of orthogonal vectors in Hilbert spaces.

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 <math>L^2([a, b])</math> converges in <math>L^2</math>-norm to some function <math>f \in L^2([a, b]).</math>  In this Note, Cauchy sequences are called "''sequences converging in the mean''" and  <math>L^2([a, b])</math> is denoted by <math>\Omega.</math> Also, convergence to a limit in <math>L^2</math>&ndash;norm is called "''convergence in the mean towards a function''". Here is the statement, translated from French:
:'''Theorem.''' ''If a sequence of functions belonging to <math>\Omega</math> converges in the mean, there exists in <math>\Omega</math> a function f towards which the sequence converges in the mean.''
Fischer goes on proving the preceding result of Riesz, as a consequence of the orthogonality of the system, and of the completeness of <math>L^2.</math>

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  
<math display="block">G_n(x) = \int_a^x g_n(t) \, \mathrm{d}t,</math>
converge uniformly on <math>[a, b]</math> to some function ''G'', continuous with bounded variation.
The existence of the limit <math>g \in L^2</math> for the Cauchy sequence is obtained by applying to ''G'' differentiation theorems from Lebesgue's theory. <br />
Riesz uses a similar reasoning in his Note, but makes no explicit mention to the completeness of <math>L^2,</math> 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''&thinsp; with bounded variation. The derivative ''f''&thinsp; of ''F'', defined almost everywhere, is square summable and has for ''Fourier coefficients'' the given coefficients.

== Completeness of ''L''<sup>''p''</sup>, &nbsp;0&nbsp;< ''p''&nbsp;&le; ∞ ==

For some authors, notably Royden,<ref>{{Cite book|title=Real analysis|last=Royden, H. L.|others=Fitzpatrick, Patrick, 1946-|date = 13 February 2017|isbn=9780134689494|edition= Fourth|location=New York, New York|oclc=964502015}}</ref> the Riesz-Fischer Theorem is the result that <math>L^p</math> is [[Complete metric space|complete]]: that every Cauchy sequence of functions in <math>L^p</math> converges to a function in <math>L^p,</math> under the metric induced by the ''p''-norm. The proof below is based on the convergence theorems for the [[Lebesgue integration|Lebesgue integral]]; the result can also be obtained for <math>p \in [1,\infty]</math> by showing that every [[Cauchy sequence]] has a rapidly converging Cauchy sub-sequence, that every Cauchy sequence with a convergent sub-sequence converges, and that every rapidly Cauchy sequence in <math>L^p</math> converges in <math>L^p.</math>

When <math>1 \leq p \leq \infty,</math> the [[Minkowski inequality]] implies that the [[Lp space]] <math>L^p</math> is a normed space. In order to prove that <math>L^p</math> is complete, i.e. that <math>L^p</math> is a [[Banach space]],  it is enough (see e.g. [[Banach space]]) to prove that every series <math>\sum u_n</math> of functions in <math>L^p(\mu)</math> such that
<math display="block">\sum \|u_n\|_p < \infty</math>
converges in the <math>L^p</math>-norm to some function <math>f \in L^p(\mu).</math> For <math>p < \infty,</math> the Minkowski inequality and the [[Monotone convergence theorem|monotone convergence theorem]] imply that
<math display="block">\int \left(\sum_{n=0}^\infty |u_n|\right)^p \, \mathrm{d}\mu \le \left(\sum_{n=0}^{\infty} \|u_n\|_p\right)^p< \infty, \ \ \text{ hence } \ \ f = \sum_{n=0}^\infty u_n</math>
is defined <math>\mu</math>&ndash;almost everywhere and <math>f \in L^p(\mu).</math> The [[Dominated convergence theorem|dominated convergence theorem]] is then used to prove that the partial sums of the series converge to ''f'' in the <math>L^p</math>-norm,
<math display="block">\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>

The case <math>0 < p < 1</math> requires some modifications, because the ''p''-norm is no longer subadditive. One starts with the stronger assumption that
<math display="block">\sum \|u_n\|_p^p < \infty</math>
and uses repeatedly that
<math display="block">\left|\sum_{k=0}^n u_k \right|^p \le \sum_{k=0}^n |u_k|^p \text{ when } p < 1</math>
The case <math>p = \infty</math> reduces to a simple question about uniform convergence outside a <math>\mu</math>-negligible set.

== See also ==

* {{annotated link|Banach space}}

==References==

{{reflist}}

* {{citation|last=Beals|first=Richard|year=2004|title=Analysis: An Introduction|location=New York|publisher=Cambridge University Press|isbn=0-521-60047-2}}.
* {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|title=Linear operators, Part I|publisher=Wiley-Interscience|year=1958}}.
* {{citation|last=Fischer|first=Ernst|title=Sur la convergence en moyenne|journal=Comptes rendus de l'Académie des sciences|volume=144|pages=1022–1024|year=1907}}.
* {{citation|last=Riesz|first=Frigyes|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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