Banach–Alaoglu and weak-star compactness

Let X be a real or complex normed space; completeness of X is not needed for this theorem. Every functional in its dual unit ball takes x into the closed scalar interval or disk of radius ∥x∥. Send the functional to this collection of values indexed by all x in X. Each coordinate set is compact, so Tychonoff’s theorem makes their product compact. The image consists of assignments that are linear and satisfy the stated coordinate bounds, and these conditions are closed under pointwise limits.

The map is injective because a functional is determined by all its evaluations. Its pulled-back product topology is precisely σ(X*,X), so the closed image proves weak-star compactness. This reasoning does not prove norm compactness: norm neighborhoods demand uniform control over every unit vector rather than finitely many chosen evaluations. It also explains why the theorem works over an uncountable index set. Replacing Tychonoff by a diagonal subsequence argument without an appropriate countable reduction would leave a genuine gap in the proof.

If X is separable, choose a countable dense set in its unit ball. On a norm-bounded subset of X*, convergence of evaluations at these vectors implies convergence at every x, because a uniform functional bound controls the approximation error. A weighted sum of the bounded evaluation discrepancies therefore gives a metric inducing weak-star topology on the dual unit ball. Compactness in this metric implies sequential compactness. Thus every bounded dual sequence has a weak-star convergent subsequence under the separable-predual assumption.

The assumption concerns X, the predual, rather than norm separability of X*. For example, ℓ¹ is separable while its dual ℓ∞ is not norm separable, yet the dual unit ball is weak-star compact and metrizable in this pairing. The metric assertion is restricted to bounded balls; it does not identify the weak-star topology on the whole dual with norm topology or claim a single countable set of tests controls unbounded functionals uniformly. In the general nonseparable setting, compactness alone provides a convergent subnet of a sequence viewed as a net, not necessarily a subsequence.

Weak-star limits preserve a dual norm bound: evaluating any fixed unit vector and passing to the limit keeps its modulus at most one. This gives a convenient way to construct bounded functionals from approximating ones. It does not make the norm differences small. Point evaluations on continuous functions illustrate the distinction: evaluation points can approach one another so every fixed continuous test agrees asymptotically, while different tests chosen for each pair still distinguish the corresponding functionals by their full dual norm.

A nonseparable predual can also make the failure of sequential compactness explicit. Give one coordinate to every subset of the natural numbers and let the nth bounded function record which subsets contain n. Any proposed subsequence can be defeated by choosing the coordinate containing exactly its even-positioned indices. The resulting scalar evaluations alternate. This does not contradict Banach–Alaoglu: its compactness is topological, and convergent subnets can have a more flexible directed index set than the natural-number subsequences that this argument excludes.