Riesz-Fischer Theorem

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.


The Lebesgue $p$-space $\map {\LL^p} \mu$, endowed with the $p$-norm $\norm {\cdot}_p$, is a Banach space.


Corollary


This page has been identified as a candidate for refactoring.
In particular: as usual
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

If a sequence $\sequence {f_k}$ in $\map {\LL^p} \mu$ converges to $f$,

then there is a subsequence $\sequence {f_{k_j}}$ that converges pointwise a.e. to $f$.


Proof

From Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach, to prove $\map {\LL^p} \mu$ is complete, it suffices to prove that every absolutely summable sequence in $\map {\LL^p} \mu$ is summable.

Let $\sequence {f_n}$ be an absolutely summable sequence in $\map {\LL^p} \mu$

Define:

$\ds \sum_{k \mathop = 1}^\infty \norm {f_k}_p =: B < \infty$

Also define:

$\ds G_n := \sum_{k \mathop = 1}^n \size {f_k}$

and:

$\ds G = \sum_{k \mathop = 1}^\infty \size {f_k}$

It is clear that the conditions of the Monotone Convergence Theorem (Measure Theory) hold, so that:

$\ds \int_X G^p = \lim_{n \mathop \to \infty} \int_X G_n^p$

By observing that:

\(\ds \norm {G_n}_p\) \(\le\) \(\ds \sum_{k \mathop = 1}^n \norm {f_n}_p\) Minkowski's Inequality/Lebesgue Spaces
\(\ds \) \(\le\) \(\ds \sum_{k \mathop = 1}^\infty \norm {f_k}_p\)
\(\ds \) \(\le\) \(\ds B\) Definition of $B$
\(\ds \) \(<\) \(\ds \infty\)

we can also say that:

$\ds \int_X \size {G_n}^p \le B^p$

and therefore:

$\ds \lim_{n \mathop \to \infty} \int_X \size {G_n}^p \le B^p$

Therefore we have that:

$\ds \int_X G^p \le B^p < \infty$

This confirms:

$G \in \map {\LL^p} \mu$

In particular:

$G \in \map{\LL^p} \mu$

entails that:

$G < \infty$ a.e.

So $\sequence {f_k}$ is absolutely summable a.e..

By Absolutely Convergent Series is Convergent/Real Numbers:

$\ds F = \sum_{k \mathop = 1}^\infty f_k$

converges a.e.


Because $\size F \le G$:

$F \in \map {\LL^p} \mu$

It only remains to show that:

$\ds \sum_{k \mathop = 1}^n f_k \to F$ in $\norm {\cdot}_p$

which we can accomplish by Lebesgue's Dominated Convergence Theorem.


This page or section has statements made on it that ought to be extracted and proved in a Theorem page.
In particular: extract the part of proof below into a page for a theorem: Dominated Pointwise Convergence implies Convergence in Lp as proved in StackExchange
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.
To discuss this page in more detail, feel free to use the talk page.


Because $\ds \size {F - \sum_{k \mathop = 1}^n f_k}^p \le (2G)^p \in \map{\LL^1}\mu$, the theorem applies.

We infer:

$\ds \norm {F - \sum_{k \mathop = 1}^n f_k}_p^p = \int_X \size {F - \sum_{k \mathop = 1}^n f_k}^p \to 0$

Therefore by Definition of Lp Norm in $\map{\LL^p}\mu$ we have that $\ds \sum_{k \mathop = 1}^\infty f_k$ converges in $\map{\LL^p}\mu$.

This shows that $\sequence {f_k}$ is summable, as we were to prove.

$\blacksquare$


Source of Name

This entry was named for Frigyes Riesz and Ernst Sigismund Fischer.


Historical Note

The Riesz-Fischer Theorem was proved jointly by Ernst Sigismund Fischer and Frigyes Riesz.

Fischer proved the result for $p = 2$, while Riesz (independently) proved it for all $p \ge 1$.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Riesz-Fischer_Theorem&oldid=689171"