Banach-Alaoglu Theorem/Proof 1
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Theorem
Let $X$ be a separable normed vector space.
Then the closed unit ball in its dual $X^*$ is weak* sequentially compact.
Proof
The aim of this proof is to show the following:
Given a bounded sequence in $X^*$, there exists a weakly convergent subsequence of that bounded sequence.
Let $\sequence {l_n}_{n \mathop \in \N}$ be a bounded sequence in $X^*$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a countable dense subset of $X$.
Choose subsequences $\N \supset \Lambda_1 \supset \Lambda_2 \supset \ldots$ such that:
- $\forall j \in \N: \map {l_k} {x_j} \to a_j =: \map l {x_j}$
as $k \to \infty$, $k \in \Lambda_j$.
Let $\Lambda$ be the diagonal sequence.
Lemma 1
$l$ can be extended to an element of $X^*$.
$\Box$
Lemma 2
$l_k \stackrel {\omega^*} {\to} l$ as $k \to \infty$, $k \in \Lambda$.
$\Box$
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Also known as
The Banach-Alaoglu Theorem is also known just as Alaoglu's Theorem.
Source of Name
This entry was named for Stefan Banach and Leonidas Alaoglu.