Monthly Archives: November 2018

Theorem. For $p\in [1, \infty)$, $L^p$ is complete. Proof. Pick a Cauchy sequence $f_n$. We can pick a subsequence $f_{n_k}$ such that $\|f_{n_{k+1}}- f_{n_k}\| \leq 2^{-k}$. Set $S_K(f) = f_{n_1} + \sum_{i=1}^{K} (f_{n_{i+1}} – f_{n_i})$ and also $S_K(g) = |f_{n_1}| … Continue reading →