Banach-Alaoglu Theorem/Proof 2

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Theorem

Let $X$ be a separable normed vector space.

Then the closed unit ball in its normed dual $X^*$ is sequentially compact with respect to the weak-$\ast$ topology.

This article needs proofreading. In particular: The separability of $X$ is stated, but not used in the exposition. So why is it stated? If $X$ does not need to be separable, it should not be. If it does need to be separable, it then needs to be shown in the course of the proof exactly what the effect of separability of $X$ being separable has on the exposition. Good point. We need to resolve the issue in Talk:Banach-Alaoglu Theorem. If you believe all issues are dealt with, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Proof

Let $X$ be a normed vector space.

Denote by $B$ the closed unit ball in $X$.

Let $X^*$ be the dual of $X$.

Denote by $B^*$ the closed unit ball in $X^*$.

Let:

$\map \FF B = \closedint {-1} 1^B$

be the topological space of functions from $B$ to $\closedint {-1} 1$.

By Tychonoff's Theorem:

$\map \FF B$

is compact with respect to the product topology.

We define the restriction map:

$R: B^* \to \map \FF B$

by:

$\forall \psi \in B^*: \map R \psi = \psi \restriction_B$

Lemma 3

$R \sqbrk {B^*}$ is a closed subset of $\map \FF B$.

$\Box$

Lemma 4

$R$ is a homeomorphism from $B^*$ with the weak* topology to its image:

$R \sqbrk {B^*}$

seen as a subset of $\map \FF B$ with the product topology.

$\Box$

Thus by Lemma 4, $B^*$ in the weak* topology is homeomorphic with $R \sqbrk {B^*}$.

This is a closed set of $\map \FF B$ (by Lemma 3) and thus compact.

By the Eberlein-Šmulian Theorem, this is sequentially compact.

This article, or a section of it, needs explaining. In particular: Exactly what the Eberlein-Šmulian Theorem that results in the above conclusion You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

There is believed to be a mistake here, possibly a typo. In particular: No, this is wrong. Eberlein-Šmulian Theorem is not sufficient You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

This needs considerable tedious hard slog to complete it. In particular: We need to use the metrization to conclude sequential compactness. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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.