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## The Riesz-Fischer Theorem

In 1907 two mathematicians, Frigyes (=Frederic'') Riesz and Ernst Fischer directed attention to a remarkable relationship between and . In spite of their different defining properties they are isomorphic and geometrically the same. Indeed, they are related by a linear transformation which is onto, one-to-one and preserves lengths and angles. This means that there is a linear mapping which takes, for example, closed right triangles in one space and maps them into closed right triangles in the other, and vica versa. See Figures 1.7 and 1.8.

Such a mapping is induced by specifying a set of elements which are orthonormal,

and which form a complete set in that they satisfy

We now know that there is a linear map

with the property that it is
1. uniquely determined by the above system of orthonormal elements,
2. is onto11, and
3. is one-to-one, i.e. the preimage of consists of only a single element.
The trail blazing contribution of Riesz and Fischer was not only to (implicitly) direct attention to this isomorphism , but also to validate its onto'' property, which presented the greatest challenge. It also was the most important one because, in order to characterize , one must not only give its formula'' but also exhibit its domain and show that it is not empty.

Riesz validated that onto'' property by showing that, given the system , the system of integral equations

has a solution belonging to implies (because of Eq.(1.7) and is implied by .

Fischer, on the other hand, validated onto'' by

(a)
considering the sequence of partial sum

(b)
observing that implies

 (19) (110)

so that the sequence is a Cauchy sequence in , and
(c)
taking advantage of the Cauchy completeness of in order to arrive at the conclusion that there exists an , namely,

corresponding to any given .

Thus both Riesz and Fischer show that to every element there corresponds an element with the numbers as its generalized Fourier coefficients. In brief, they show that , Eq.(1.8), is an onto map.

That is one-to-one does not require their hard work. That property follows directly from Eq.(1.11) according to which whenever .

In summary, one has the following

Riesz-Fischer Theorem

Given: (i)
An orthonormal system in the (Cauchy) complete
(ii)
A sequence of numbers such that

Conclusion:
There exists an element with as its generalized Fourier coefficient, i.e. such that

where

#### Footnotes

...onto11
This claim is often also stated by saying that the preimage of , namely, the set of elements

is non-empty.

Next: Orthogonal Basis and Orthogonalization Up: Hilbert Spaces Previous: Two Prototypical Examples   Contents   Index
Ulrich Gerlach 2010-12-09