Talk:Riesz-Fischer Theorem

Added a corollary

It is Corollary 7.11. in this note. The corollary follows immediately from the proof.

To explain that we must take a subsequence in the corollary (the full sequence need not converge pointwise a.e.), we can add an example of Converge in $L^p$, but not converge pointwise a.e. for $1\le p<\infty$. There is a StackExchange question Does convergence in Lp imply convergence almost everywhere?

Consider the "typewriter sequence" defined by the formula:

$f_n:=1_{\left[\frac{n-2^k}{2^k},\frac{n-2^k+1}{2^k}\right]}$

where $k$ is an integer such that $2^k\leq n<2^{k+1}$, the sequence converges to zero in $L^p$ norm, but not pointwise.

--Hbghlyj (talk) 14:04, 19 March 2024 (UTC)

Include $p = \infty$?

This page only proves Riesz-Fischer Theorem for $p\in\R$, but in Theorem 7.10 (Riesz-Fischer theorem) of this note also includes $p = \infty$.

There is a second part in the proof: $\text { Second, suppose that } p=\infty \text {. If }\left\{f_k\right\} \text { is Cauchy in } L^{\infty} \text {, then }$ --Hbghlyj (talk) 14:40, 19 March 2024 (UTC)