# Riesz-Fischer Theorem

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## 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, is a complete metric space.

## Proof

## 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

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