User:Addem/riesz fischer

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
To show that $\map {\LL^p} \mu$ is complete we consider an arbitrary norm Cauchy sequence $\{f_n\}$ taken from $\map {\LL^p} \mu$, and demonstrate that there is a function in $\map{\LL^p}\mu$ to which $\{f_n\}$ converges in $\map{\LL^p}\mu$. However, it suffices to show this merely for a subsequence, since every Cauchy sequence with a convergent subsequence must converge. Therefore we consider a subsequence $\{f_{n_k}\}$ such that $\norm {f_{n_k}-f_{n_{k-1}}}_p<1/2^k$, which must exist because $\{f_n\}$ is Cauchy.

First we show that $\{f_{n_k}\}$ converges. Define $g_m = |f_{n_1}|+\sum_{k=1}^m|f_{n_k}-f_{n_{k-1}}|$