Riesz-Fischer theorem: completeness of $L^p$

Let $(X,\mathcal{A},\mu )$ be a measure space and $1\le p\le \mathrm{\infty }$. Write $L^p(\mu )$ for the space of measurable $f$ with $\mathrm{\parallel }f{\mathrm{\parallel }}_p<\mathrm{\infty }$, modulo equality almost everywhere.

Riesz-Fischer theorem. $(L^p(\mu ),\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p)$ is a Banach space: every $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p$-Cauchy sequence converges in $L^p(\mu )$. Moreover, if $f_n\to f$ in $L^p(\mu )$ then some subsequence converges to $f$ pointwise almost everywhere.

For $p=2$ this makes $L^2(\mu )$ a Hilbert space, and the classical 1907 form of the theorem is the surjectivity half of the correspondence with ${\mathrm{\ell }}^2$: for an orthonormal system $(e_n)$ in $L^2$ and any $(c_n)\in {\mathrm{\ell }}^2$ the series ${\sum }_n{c}_n{e}_n$ converges in $L^2$, so every square-summable sequence is the sequence of Fourier coefficients of an $L^2$ function.

Not proved in this library. It is recorded with citations and used in no proof here.

What would prove it. Minkowski's inequality (Holder and Minkowski inequalities in integral form ‡) to know $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p$ is a norm, then the standard criterion that a normed space is complete when every absolutely convergent series converges, applied through the monotone convergence theorem (Monotone convergence theorem (Beppo Levi) ‡) to ${\sum }_n\mathrm{\parallel }{f}_{n+1}-f_n{\mathrm{\parallel }}_p<\mathrm{\infty }$, with the dominated convergence theorem (Dominated convergence theorem ‡) identifying the limit. The subsequence statement falls out of the same construction. The case $p=\mathrm{\infty }$ is separate and easier, since a countable union of null sets is again null.

Which page it serves. The completeness page, which proves $\mathbb{R}$ and ${\mathbb{R}}^n$ complete and the bounded functions with the sup metric complete, and the function-space page, which proves $C([0,1],\mathbb{R})$ complete for the uniform metric. The $L^1$ metric on $C[a,b]$ is not built here at all, which is the honest motivation for the Lebesgue theory: the completion of $C[a,b]$ under the $L^1$ metric exists abstractly, and the theorem above says it is a space of functions, namely $L^1[a,b]$. Without measure theory the completion stays an abstract object with no description.

Naming. The name covers both the completeness theorem for $L^p$ and the 1907 result about Fourier coefficients in $L^2$, which were proved independently by F. Riesz and E. Fischer. This library keeps both under this id because they are the same theorem in the case $p=2$.