Banach–Alaoglu theorem - Imported Knowlpedia Content

Statement

Let $X$ be a normed vector space over $\mathbb R$ or $\mathbb C$ , and let $X'$ be its continuous dual. The Banach–Alaoglu theorem states that the closed dual unit ball

B_{X'}=\{\varphi\in X':\lVert\varphi\rVert\leq1\}

is compact in the weak-star topology $\sigma(X',X)$ . Completeness of $X$ is not required. The conclusion is weak-star compactness; in an infinite-dimensional setting, the same ball is not compact in its norm topology Conway, Chapter V.

Proof mechanism

Evaluation embeds $B_{X'}$ into the product

\prod_{x\in X}\{z:|z|\leq\lVert x\rVert\}.

Each factor is compact, so the product is compact by Tychonoff's theorem. The linearity and norm inequalities defining the image are closed conditions in the product topology. The subspace topology induced by this embedding is exactly $\sigma(X',X)$ , proving compactness.

Consequences and scope

If $X$ is separable, the weak-star topology on $B_{X'}$ is metrizable; Banach–Alaoglu then supplies sequential compactness as well. Without separability, compactness need not be detectable by sequences. Applying the theorem to the bidual, together with the canonical embedding, helps show that the closed unit ball of a reflexive Banach space is weakly compact Rudin, Chapter 3.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. Springer DOI record. Relevant: Chapter V, “Weak Topologies.”
Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: Chapter 3, compactness in dual spaces.