[Math] Bounded sequence in Hilbert space contains weak convergent subsequence

Question

In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence.

Is there any short proof? Thanks a lot.

Answer 1

Suppose $M$ bounds the sequence. Then, if we think of $H$ as sitting inside $H^{**}$, then for any $T \in H^\ast$ with $\|T\| \leq 1$, we have $\|x_n(T)\| = \|Tx_n\| \leq \|x_n\| \leq M$, so the operator norms of the $x_n$ thought of as operators on $H^\ast$ are bounded by $M$. Apply Banach-Alaoglu. (To the unit ball of $H^{\ast \ast}$.)