Weak-star topology

The weak-star topology $\sigma (X^{\ast },X)$ is pointwise convergence of functionals on $X$. If $X$ is nonreflexive, it is strictly weaker than the weak topology $\sigma (X^{\ast },X^{\ast \ast })$ on $X^{\ast }$.

The continuous dual space of ${\ell }^1$ is ${\ell }^{\infty }$ under the complex-linear coordinate pairing. Its subspace $c_0$ is norm closed: uniform approximation by a sequence with a small tail leaves a small tail. Nevertheless its preannihilator in ${\ell }^1$ is zero, because the coordinate unit vectors belong to $c_0$ and test every coordinate. Its double continuous annihilator is therefore all of ${\ell }^{\infty }$. It is weak-star dense directly as well: the finite truncations of any bounded sequence lie in $c_0$ and converge against every ${\ell }^1$ test by absolute summability. This demonstrates why norm closure cannot replace closure in the weak-star topology in the dual-side annihilator theorem.

The same Banach space can acquire incomparable weak-star topologies from different isometric preduals. Pair ${\ell }^1$ with $c_0$ by $\sum {\alpha }_n{\beta }_n$ and with the space $c$ of convergent sequences by ${\alpha }_1{\gamma }_{\infty }+{\sum }_{n\ge 2}{\alpha }_n{\gamma }_{n-1}$. The functional $\alpha \mapsto {\alpha }_1$ is continuous only for the first topology: $e_1-e_n\to 0$ for the second but its values remain one. Conversely $\alpha \mapsto \sum {\alpha }_n$ is continuous only for the second: $e_n\to 0$ for the first but its values remain one. More generally, comparable weak-star topologies arising from isometric preduals agree on the common closed unit ball, since its identity map is a continuous bijection from a compact space to a Hausdorff space by the Banach-Alaoglu theorem.

A bounded measurable one-periodic function rescaled to frequency $j$ converges weak-star in $L^{\infty }(\mathbb{R})$ to its period average. Subtract the average and take a bounded periodic primitive $F$. Its integral on $[a,b]$ after rescaling is $[F(jb)-F(ja)]/j$, which tends to zero. Sequential Banach-Alaoglu with separable predual $L^1$ then identifies every cluster point by interval integrals; uniqueness of all cluster points gives full weak-star convergence. Localizing to bounded intervals avoids falsely assuming that a bounded cluster function is globally integrable.

If $T=S^{\ast }$ for a bounded operator on a separable Banach space, and $C$ is a nonempty weak-star closed convex subset of the dual unit ball with $T(C)\subseteq C$, then $T$ has a fixed point in $C$. The Cesàro averages of the orbit remain in $C$, and $T$ minus the identity sends those averages to vectors of norm at most $2/n$. Weak-star compactness supplies a cluster point, and the adjoint action is weak-star continuous. The corresponding assertion with only weak closedness is false for the right shift on the probability simplex in ${\ell }^1$, the dual of $c_0$.

The continuous dual of a separable Banach space is separable in its weak-star topology, even when it is not norm separable. Banach-Alaoglu theorem and weak-star metrizability of the dual ball make the dual unit ball a separable compact metric space. The countable union of positive integer multiples of a countable dense subset of that ball is dense in the whole dual. This does not imply global weak-star metrizability.

For an infinite-dimensional normed space, the weak-star open unit ball of its dual is not a Baire space. The smaller closed norm balls are weak-star closed and have empty relative interior: a nonzero functional annihilating any finitely many tested vectors can move the norm beyond the smaller radius while remaining below one. Their countable union is the open norm ball. Thus this topology is not topologically complete, regardless of separability.

A countable local base at zero would give countably many finite sets of evaluation vectors. Any other evaluation must lie in the span of one of those finite sets: otherwise a bounded functional vanishing on that set but not on the evaluation vector, supplied by Hahn-Banach theorem, contradicts inclusion of neighbourhoods after scalar rescaling. Thus the predual has countable Hamel dimension, impossible for an infinite-dimensional Banach space by the Baire category theorem. This concerns the entire dual, unlike weak-star metrizability of the dual ball.

A linear functional on $X^{\ast }$ is continuous for the weak-star topology exactly when it is evaluation at an element of $X$. Continuity makes it vanish on the common kernel of finitely many evaluations, so it is a linear combination of those evaluations. The canonical embedding into the bidual identifies this dual with $X$.

In $L^{\infty }(\mathbb{R})=(L^1(\mathbb{R}){)}^{\ast }$, the sequence $f_n(x)=\sin (nx)$ converges weak-star to zero by the Riemann-Lebesgue lemma, while $f_n^2$ converges weak-star to the constant $1/2$. Nonlinear pointwise operations therefore need not preserve weak-star limits.

If $X$ is separable, the weak-star topology on $B_{X^{\ast }}$ is metrizable. For a dense sequence $(x_n)$ in $B_X$, a compatible metric is obtained by summing bounded multiples of $2^{-n}|f(x_n)-g(x_n)|$.

The dual unit ball $B_{X^{\ast }}$ is weak-star metrizable if and only if $X$ is separable. For the converse direction, Banach-Alaoglu theorem makes a metrizable dual ball a compact metric space, and the evaluation map embeds $X$ isometrically into the separable space $C(B_{X^{\ast }})$.

A Banach space $X$ is a Grothendieck space when every weak-star convergent sequence in $X^{\ast }$ is weakly convergent. Every reflexive Banach space has this property, and every separable Grothendieck space is reflexive.

For a weak-star compact set $K\subseteq X^{\ast }$, its Szlenk derivation at scale $\epsilon >0$ isIt removes points having a relatively weak-star open neighbourhood of norm diameter at most $\epsilon$.

Transfinite iteration of this operation defines the Szlenk index; one derivation step is not itself the index.

If $X^{\ast }$ is separable and $K$ is a nonempty weak-star closed subset of $B_{X^{\ast }}$, then $K_{\epsilon }^{\prime }$ is a proper subset of $K$ for every $\epsilon >0$. Metrizability produces sequences witnessing membership in the derivative, while a Baire-category argument applied to a universal weakly null sequence yields the strict inclusion.