Theorem by Fischer-Riesz

The Riesz-Fischer theorem is a statement from the functional analysis . Ernst Sigismund Fischer and Frigyes Riesz independently proved this proposition in 1907. It is for this reason that the statement bears their names. In the literature today there are different sentences which bear her name and which are in part generalizations of this sentence.

Classic movement by Fischer-Riesz

Fischer and Riesz proved the following statement. The space of the square-integrable functions is isometrically isomorphic to the sequence space of the square-summable functions ${\displaystyle L^{2}([0,1])}$ ${\displaystyle \ell ^{2}(\mathbb {N} )}$

${\displaystyle L^{2}([0,1])\cong \ell ^{2}(\mathbb {N} ).}$

This can also be formulated less abstractly in the language of real analysis . So a measurable function is in if and only if its Fourier series converges with respect to the norm. In the following, the -space is formed by the interval , this saves normalization, but the statement is also correct for all other compact intervals. ${\displaystyle L^{2}([-\pi ,\pi ])}$ ${\displaystyle L^{2}}$ ${\displaystyle L^{2}}$ ${\displaystyle [-\pi ,\pi ]}$

The Fourier series broken off at the -th term is a function that can be integrated into the square ${\displaystyle N}$ ${\displaystyle f}$

${\displaystyle {\mathcal {F}}_{N}(f)(x)=\sum _{n=-N}^{N}a_{n}\,\mathrm {e} ^{inx},}$

where the nth coefficient of the series is passed through ${\displaystyle a_{n}}$

${\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} x}$

given is. For a square-integrable function then holds ${\displaystyle f}$

${\displaystyle \lim _{N\to \infty }\left\Vert {\mathcal {F}}_{N}(f)-f\right\|_{L^{2}}=\lim _{N\to \infty }\left\Vert \sum _{n=-N}^{N}a_{n}\,\mathrm {e} ^{in\cdot }-f\right\|_{L^{2}}=0.}$

The isomorphism between and is the transformation into a Fourier series. ${\displaystyle L^{2}}$ ${\displaystyle \ell ^{2}(\mathbb {N} )}$

Generalized theorem from Fischer-Riesz

The following, more general statement can often be found under the name of the sentence by Fischer-Riesz.

statement

If a Hilbert space and an orthonormal basis of , then the map is ${\displaystyle H}$ ${\displaystyle \left(e_{i}\right)_{i\in I}}$ ${\displaystyle H}$

${\displaystyle \Phi \colon \,H\to \ell ^{2}(I);\quad x\mapsto \left(\langle x,e_{i}\rangle \right)_{i\in I}}$

an isometric isomorphism .

Inferences

• Let and be two matching index sets. Two Hilbert spaces and with orthonormal bases and are isometrically isomorphic if and have the same cardinality . ${\displaystyle I}$ ${\displaystyle J}$ ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle \left(e_{i}\right)_{i\in I}}$ ${\displaystyle \left(e_{j}\right)_{j\in J}}$ ${\displaystyle I}$ ${\displaystyle J}$

• Every orthonormal system in a Hilbert space can be supplemented to an orthonormal basis (which follows directly from Zorn's Lemma ), in particular every Hilbert space has an orthonormal basis, since the empty set is always an orthonormal system . Thus, according to Fischer-Riesz's theorem, every Hilbert space is isomorphic to space . ${\displaystyle B}$ ${\displaystyle \ell ^{2}(B)}$

• In other words, the full sub-category of spaces for arbitrary sets in the Hilbert space category with suitable morphisms (linear operators, bounded linear operators, linear contractions) is equivalent to this. ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle I}$

• From the theorem it can be concluded that every separable infinite-dimensional Hilbert space is isometrically isomorphic to the sequence space . ${\displaystyle \ell ^{2}(\mathbb {N} )}$

Completeness of the L ^p spaces

→ Main article : Duality of spaces ${\displaystyle L^{p}}$

The statement that the -rooms for with the norm ${\displaystyle L^{p}(\Omega ,\mu )}$ ${\displaystyle 1\leq p\leq \infty }$

${\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}$

Banach spaces , i.e. especially complete , is often referred to as the Fischer-Riesz theorem.

For the case and as a Lebesgue measure , this follows from the proof of the (classical) Fischer-Riesz theorem. So the sequence converges in if and only if is a function. ${\displaystyle p=2}$ ${\displaystyle \mu }$ ${\displaystyle \textstyle a_{n}:=\int _{\Omega }f(x)\,\mathrm {e} ^{-inx}\,\mathrm {d} \mu (x)}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle f}$ ${\displaystyle L^{2}}$

For example , the completeness of the space results from its reflexivity , which results from the duality of Lp spaces . Every reflexive normalized space is a Banach space, because by definition it is isomorphic to the complete bidual space . ${\displaystyle 1<p<\infty }$ ${\displaystyle L^{p}}$

Individual evidence

1. ↑ Sur les systèmes orthogonaux de fonctions, CR Paris 144 (1907) 615-619

literature

• Dirk Werner : functional analysis , Springer-Verlag, Berlin, 2007, ISBN 978-3-540-72533-6
• H. Heuser: functional analysis , Teubner-Verlag, 2006, ISBN 3-8351-0026-2 , contains historical comments