The Riesz-Fischer Theorem

In 1907 two mathematicians, Frigyes (=``Frederic'') Riesz and Ernst Fischer directed attention to a remarkable relationship between $\ell ^2$ and $L^2(a,b)$ . 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 $u_1, u_2, \cdots, u_k, \cdots$ which are orthonormal,

$$\langle u_k,u_\ell\rangle =\delta_{k\ell}~,$$

and which form a complete set in that they satisfy

$$\sum_{k=1}^\infty \vert\langle u_k,f\rangle\vert^2=\Vert f \Vert^2 ~\textrm{whenever}~ f\in L^2~.$$ (17)

We now know that there is a linear map

$$\begin{align*}\begin{array}{rcl} ~&\mathcal{F}&~\\ L^2(a,b) &\longrightarrow&\ell^2\\ f & \sim\leadsto&\mathcal{F} [f]=\{c_k\} \end{array}\end{align*}$$ (18)

with the property that it is

1. uniquely determined by the above system $\{ u_k\}$ of orthonormal elements,
2. is onto¹¹, and
3. is one-to-one, i.e. the preimage $\mathcal{F}^{-1}\left[\{ c_k\} \right]$ of $\{ c_k\}$ consists of only a single element.

The trail blazing contribution of Riesz and Fischer was not only to (implicitly) direct attention to this isomorphism $\mathcal{F}$ , but also to validate its ``onto'' property, which presented the greatest challenge. It also was the most important one because, in order to characterize $\mathcal{F}$ , 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 $\{ u_k\}$ , the system of integral equations

$$\int_a^b u_k(x) f(x) \rho(x)\, dx=c_k~~(k=1,2,3,\cdots )$$

has a solution $f$ belonging to $L^2(a,b)$ implies (because of Eq.(1.7) and is implied by $\sum_{k=1}^\infty \vert c_k\vert^2<\infty$ .

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

(a)

considering the sequence $f_1,~f_2,\cdots,~ f_N, \cdots$ of partial sum

$$f_N=\sum_{k=1}^N c_ku_k$$

(b)

observing that $\sum_{k=1}^\infty \vert c_k\vert^2<\infty$ implies

$$\Vert f_{N+p} -f_N \Vert^2 =\Vert c_{N+1} u_{N+1}+\cdots+c_{N+p}u_{N+p}\Vert^2$$ (19)

$$=\sum_{k=N+1}^{N+p} \vert c_k\vert^2 \longrightarrow 0~\textrm{as}~ N\longrightarrow \infty$$ (110)

so that the sequence $f_1,~f_2,\cdots,~ f_N, \cdots$ is a Cauchy sequence in $L^2(a,b)$ , and

(c)

taking advantage of the Cauchy completeness of $L^2(a,b)$ in order to arrive at the conclusion that there exists an $f\in L^2(a,b)$ , namely,

$$f=\sum_{k=1}^\infty c_ku_k~,$$ (111)

corresponding to any given $\{ c_k \}\in \ell^2$ .

Thus both Riesz and Fischer show that to every element $\{ c_1,c_2,\cdots,c_k,\cdots\} \in \ell^2$ there corresponds an element $f\in L^2(a,b)$ with the numbers $c_1,c_2,\cdots,c_k,\cdots$ as its generalized Fourier coefficients. In brief, they show that $\mathcal{F}$ , Eq.(1.8), is an onto map.

That $\mathcal{F}$ is one-to-one does not require their hard work. That property follows directly from Eq.(1.11) according to which $f=0$ whenever $\{ c_k \}=\{ 0,\cdots,0\}$ .

In summary, one has the following

Riesz-Fischer Theorem

Given: (i)

An orthonormal system $\{ u_k\}$ in the (Cauchy) complete $L^2(a,b)$

(ii)

A sequence of numbers $c_1,c_2,\dots,c_k,\cdots$ such that

$$\sum_{k=1}^\infty \vert c_k\vert^2<\infty~.$$

Conclusion:

There exists an element $f\in L^2(a,b)$ with $c_1,c_2,\dots,c_k,\cdots$ as its generalized Fourier coefficient, i.e. such that

$$\sum_{k=1}^\infty \vert c_k\vert^2=\Vert f\Vert^2 ~~(\Leftrightarrow \lim_{N\to\infty} \Vert f-\sum_{k=1}^N c_ku_k\Vert^2=0 ~!)$$

where

$$c_k=\langle u_k,f\rangle,~~(k=1,2,\cdots )$$

Footnotes

...onto¹¹

This claim is often also stated by saying that the preimage $\mathcal{F}^{-1}\left[\{ c_k\} \right]$ of $\{ c_k\}$ , namely, the set of elements

$$\mathcal{F}^{-1}\left[\{ c_k\} \right]=\left\{ f:\mathcal{F}[f]=\{ c_k\}\right\}~,$$

is non-empty.