Banach-Alaoglu is the basic compactness theorem for dual balls. It replaces missing norm compactness in infinite-dimensional spaces by compactness for pointwise convergence on the predual.
Embedding into a product
Weak-star compactness is easiest to see after embedding a dual ball into a product of scalar compact sets. The coordinates of this product are evaluations at points of the original space.
Definition. For a normed space \(X\), let \(B_{X^*}=\{f\in X^*:\|f\|\le1\}\). The weak-star topology is the topology of pointwise convergence on \(X\).
Lemma. The map \(\Phi:B_{X^*}\to\prod_{x\in X}\{z\in\mathbb F:|z|\le\|x\|\}\), \(\Phi(f)=(f(x))_{x\in X}\), is injective and identifies the weak-star topology with the subspace topology inherited from the product.
Proof. Injectivity holds because a functional is determined by its values on all \(x\in X\). Product convergence is pointwise convergence of coordinates, and the coordinate indexed by \(x\) is exactly the evaluation map \(f\mapsto f(x)\). Thus the initial topology generated by evaluations is the subspace topology through \(\Phi\).
Banach-Alaoglu
The theorem is a compactness statement for all bounded functionals at once. It uses Tychonoff compactness, then verifies that the algebraic and norm conditions are closed coordinate conditions.
Theorem. The closed unit ball \(B_{X^*}\) is compact in the weak-star topology.
Proof. Each coordinate set \(\{z:|z|\le\|x\|\}\) is compact in \(\mathbb F\), so their product is compact by Tychonoff's theorem. It remains to show that \(\Phi(B_{X^*})\) is closed in the product.
Let a net \(\Phi(f_\alpha)\) converge pointwise to \(a=(a_x)\). Pointwise limits preserve linearity: \(a_{x+y}=\lim f_\alpha(x+y)=a_x+a_y\) and \(a_{\lambda x}=\lambda a_x\). Also \(|a_x|\le\|x\|\). Hence \(f(x)=a_x\) defines a linear functional with \(\|f\|\le1\). Therefore \(a=\Phi(f)\), so the image is closed and compact.
Weak compactness in reflexive spaces
In reflexive spaces, weak compactness of the closed unit ball is inherited from weak-star compactness of the double dual ball.
Corollary. If \(X\) is reflexive, then \(B_X\) is compact in the weak topology.
Proof. Under the canonical isometric isomorphism \(J_X:X\to X^{**}\), the weak topology on \(X\) corresponds to the weak-star topology \(\sigma(X^{**},X^*)\) on \(J_X(X)\). Since \(J_X(B_X)=B_{X^{**}}\) when \(X\) is reflexive, Banach-Alaoglu gives weak compactness of \(B_X\).
Metrizability on separable preduals
Weak-star compactness is not usually sequential. Separability of the predual gives a countable set of coordinates, and on bounded sets this makes the topology metrizable.
Proposition. If \(X\) is separable, the weak-star topology on \(B_{X^*}\) is metrizable. If \((x_n)\) is dense in \(B_X\), one compatible metric is \(d(f,g)=\sum_{n=1}^\infty 2^{-n}|f(x_n)-g(x_n)|/(1+|f(x_n)-g(x_n)|)\).
Proof. The metric topology is generated by the countable evaluation maps \(f\mapsto f(x_n)\). It is weaker than the weak-star topology. Conversely, if \(f_k\to f\) in this metric and \(x\in B_X\), choose \(x_n\) with \(\|x-x_n\|<\varepsilon\). Since \(f_k,f\) all have norm at most one, \(|(f_k-f)(x)|\le |(f_k-f)(x_n)|+2\varepsilon\). Let \(k\to\infty\) and then \(\varepsilon\downarrow0\). Thus all evaluations converge.
Weak-star sequential compactness
Once a compact space is metrizable, compactness can be read by sequences. This is the form most useful in analysis, statistics, and variational arguments.
Corollary. If \(X\) is separable, every sequence in \(B_{X^*}\) has a weak-star convergent subsequence.
Proof. By Banach-Alaoglu, \(B_{X^*}\) is weak-star compact. By the metrizability proposition, compactness is equivalent to sequential compactness on this ball.
Statistical reading
For statistical applications, weak-star compactness is the functional-analytic form of tightness-like extraction: bounded linear summaries have convergent subnet or subsequence limits under the right separability assumptions.
Example. When \(X=L^1(\mu)\), the dual \(X^*=L^\infty(\mu)\) has weak-star compact unit ball. Bounded likelihood ratios or bounded estimating functions often live naturally in such weak-star compact sets, where pointwise testing against \(L^1\) functions replaces norm compactness.