- Suppose \(\{ f _ { n } \} _ { n = 1 } ^ { \infty } \subset L ^ { 2 } ( \mathbb { R } )\) is a sequence that converges to 0 in the \(L ^ { 2 }\) norm; in other words,
\[ | | f _ { n } | | _ { L ^ { 2 } ( \mathbb { R } ) } = \left( \int _ { - \infty } ^ { \infty } | f _ { n } | ^ { 2 } \ d x \right) ^ { \frac { 1 } { 2 } } \to 0 . \]
Prove that there exists a subsequence \(\{ f _ { n _ { k } } \}\) such that \(f _ { n _ { k } } \to 0\) almost everywhere.
Solution
1. There is a subsequence with \(\sum_k\int\abs{f_{n_k}}^2<\infty\).
Since \(\norm{f_n}_2\to0\), choose \(n_1<n_2<\cdots\) with \(\norm{f_{n_k}}_2\le2^{-k}\); then \(\sum_k\norm{f_{n_k}}_2^2\le\sum_k4^{-k}<\infty\).
2. Q.E.D.
By the monotone convergence theorem and step 1, \(\int\sum_k\abs{f_{n_k}}^2=\sum_k\int\abs{f_{n_k}}^2<\infty\), so \(\sum_k\abs{f_{n_k}(x)}^2<\infty\) for almost every \(x\). For such \(x\) the terms tend to \(0\), so \(f_{n_k}(x)\to0\).