Riesz–Fischer theorem

Remarks

The proof for $p < \infty$ starts from a Cauchy sequence $(f_n)$ and picks a subsequence with $\norm{f_{n_{k+1}} - f_{n_k}}_p < 2^{-k}$. The function $g = \abs{f_{n_1}} + \sum_k \abs{f_{n_{k+1}} - f_{n_k}}$ lies in $L^p$ by Minkowski's inequality and the monotone convergence theorem. So the subsequence converges almost everywhere to a function $f$, and Fatou's lemma shows that $f \in L^p$ and $f_n \to f$ in $L^p$. A by-product: for $p < \infty$, every sequence that converges in $L^p$ has a subsequence that converges almost everywhere. The case $p = \infty$ is a direct argument with uniform convergence outside a null set.

The original form concerns $p = 2$. If $(\varphi_n)$ is an orthonormal sequence in $L^2(\mu)$ and $(c_n)$ is a sequence of scalars with $\sum_n \abs{c_n}^2 < \infty$, then there is $f \in L^2(\mu)$ with $\ip{f}{\varphi_n} = c_n$ for all $n$. For the trigonometric system on $[0, 2\pi]$, every square-summable sequence is the sequence of Fourier coefficients of some square-integrable function.

The Lebesgue integral is essential here. The continuous functions on $[0, 1]$ with the norm $\int_0^1 \abs{f}\, dx$ do not form a complete space.

Going deeper

For a student

Completeness makes the tools of Banach space theory available on $L^p$: the Baire category theorem, the open mapping theorem and the uniform boundedness principle. For $p = 2$ it makes $L^2(\mu)$ a Hilbert space, which is the setting of Fourier series and of quantum mechanics. The proof also gives the useful fact that $L^p$ convergence implies almost everywhere convergence along a subsequence, but not for the whole sequence. The "typewriter" sequence of indicator functions of shrinking intervals that sweep across $[0, 1]$ converges to $0$ in $L^1$ but at no point.