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
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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): $12.7$