# Riesz–Fischer theorem

Riesz–Fischer theorem For more theorems that are sometimes called Riesz's theorem, see Riesz theorem.

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces {displaystyle L^{p}} from Lebesgue integration theory are complete.

Contents 1 Modern forms of the theorem 2 Example 3 History: the Note of Riesz and the Note of Fischer (1907) 4 Completeness of Lp,  0 < p ≤ ∞ 5 See also 6 References Modern forms of the theorem The most common form of the theorem states that a measurable function on {displaystyle [-pi ,pi ]} is square integrable if and only if the corresponding Fourier series converges in the Lp space {displaystyle L^{2}.} This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by {displaystyle S_{N}f(x)=sum _{n=-N}^{N}F_{n},mathrm {e} ^{inx},} where {displaystyle F_{n},} the nth Fourier coefficient, is given by {displaystyle F_{n}={frac {1}{2pi }}int _{-pi }^{pi }f(x),mathrm {e} ^{-inx},mathrm {d} x,} then {displaystyle lim _{Nto infty }leftVert S_{N}f-fright|_{2}=0,} where {displaystyle |,cdot ,|_{2}} is the {displaystyle L^{2}} -norm. Conversely, if {displaystyle {a_{n}},} is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that {displaystyle sum _{n=-infty }^{infty }left|a_{n}rightvert ^{2}0,} there exists a finite set {displaystyle B_{0}} in A such that {displaystyle |x-sum _{yin B}langle x,yrangle y|n}|u_{ell }|right)^{p},mathrm {d} mu rightarrow 0{text{ as }}nrightarrow infty .} The case {displaystyle 0

Si quieres conocer otros artículos parecidos a Riesz–Fischer theorem puedes visitar la categoría Fourier series.

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