The Riesz-Fischer Theorem

What guarantee do we have that $L^2$ spaces are Cauchy complete? The answer is given by the

Riesz-Fischer Theorem

Given: Let $\{ f_1,f_2,\dots\}$ be a sequence in $L^2(a,b)$ , i.e., a sequence of square integrable functions.

Let $\lim\limits_{k,p\to\infty} \Vert f_k-f_p\Vert^2\equiv\lim\limits_{k,p\to \infty}\int^b_a\vert f_k(x)-f_p(x)\vert^2 \rho (x)\,dx=0$ , i.e., $\{ f_1,f_2,\dots\}$ is a Cauchy sequence with respect to $\Vert\cdots\Vert$ .

Conclusion: $\exists$ a square integrable function $f(x)$ (i.e., $f\in L^2(a,b)$ ) such that

$$\lim_{n\to\infty}\int^b_a \vert f-f_n\vert^2\rho (x)\,dx = 0$$

i.e., $\{ f_n\}$ converges ``in the mean'' to $f\in L^2(a,b)$ , i.e., $L^2(a,b)$ is Cauchy complete.

Roughly speaking, square integrability gives us two for the price of one: (i) Not only are we guaranteed the existence of a square integrable limit of any Cauchy sequence of square integrable functions, but (ii) we are also guaranteed that one has closure under addition; in particular if we add that limit to any other square integrable function we get another square integrable function.

Summary: $L^2(a,b)$ is a Cauchy complete inner product space, i.e.,

$$L^2(a,b)~\textrm{is~a~Hilbert~space}\,.$$