Riesz–Fischer theorem

The set of Fischer- Riesz is a statement from the functional analysis. Proved Ernst Sigismund Fischer and Riesz Frigyes in 1907 independently this sentence. For this reason, the statement bears their name. In the literature there are different sets today that bear their name and in part are generalizations of this theorem.

Classic set of Fischer- Riesz

Fischer and Riesz proved the following statement. The space of square - integrable functions is isometrically isomorphic to the sequence space of square - summable functions so

This can be also less abstract formulation in the language of real analysis. For a measurable function is exactly then when its Fourier series converges with respect to the norm. In the following, the space formed by the interval, this will save normalizations, but the statement is true for all other compact intervals.

Is the -th element broken Fourier series of a square - integrable function

Wherein the n- th coefficient series is represented by

Is given. Therefore applies to a square - integrable function then

The isomorphism between and is thus the transformation into a Fourier series.

Generalized set of Fischer- Riesz

Often you will also find the following, more general statement under the name of Fischer- Riesz theorem.

Statement

Is a Hilbert space and an orthonormal basis of, as is the mapping

An isometric isomorphism.

Conclusions

• Let and be two matching index sets. Two Hilbert spaces with orthonormal bases and are isometrically isomorphic if and only have the same cardinality.

• Each orthonormal system in a Hilbert space can be extended to an orthonormal basis ( which directly follows from the Lemma of Zorn), in particular, each Hilbert space, since the empty set is always an orthonormal system, an orthonormal basis. Thus, by the theorem of Fischer- Riesz every Hilbert space is isomorphic to the space.

• In other words, the full subcategory of spaces for arbitrary sets in the category of Hilbert spaces with suitable morphisms ( linear operators, bounded linear operators, linear contractions ) is equivalent to this.

• From the theorem we conclude that every separable infinite-dimensional Hilbert space to the result space is isometrically isomorphic.

Completeness of Lp- spaces

The statement that the rooms for the standard

Banach spaces thus in particular are complete, is also often referred to as a set of Fischer- Riesz.

In the event of and as Lebesgue measure this, it follows from the proof of (classical) set of Fischer- Riesz. So the sequence converges if and only in when a function is.

For the completeness of the space is - for example, because of its reflexivity, which results from the duality of Lp- spaces. Each reflexive normed space is a Banach space, because it is by definition isomorphic to the full Bidualraum.