# Riesz-Fischer Theorem

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## Contents

## Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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

The Lebesgue $p$-space $\mathcal L^p \left({\mu}\right)$, endowed with the $p$-norm, is a complete metric space.

## Proof

## Source of Name

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

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

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $12.7$