Weak-star topology - Knowlpedia

Let $E$ be a topological vector space, and denote its topological dual by $E'$ . The weak-star topology $\sigma(E',E)$ is the weakest topology on $E'$ for which every evaluation map

\operatorname{ev}_x:E'\to\mathbb K,\qquad \varphi\mapsto\varphi(x), \quad x\in E,

is continuous. Equivalently, it is the topology generated by the seminorms $p_x(\varphi)=|\varphi(x)|$ . Hence a net $(\varphi_i)$ converges weak-star to $\varphi$ exactly when $\varphi_i(x)\to\varphi(x)$ for every $x\in E$ .

Neighborhoods and continuity

A basic neighborhood of $0$ specifies finitely many vectors $x_1,\ldots,x_m\in E$ and tolerances $\varepsilon_j>0$ :

\{\varphi\in E':|\varphi(x_j)|<\varepsilon_j \text{ for }1\leq j\leq m\}.

Thus weak-star convergence tests one vector at a time; it does not require uniform convergence on the unit ball or on bounded subsets of $E$ .

Dependence on the predual

The notation $\sigma(E',E)$ records both sides of the dual pairing. If a Banach space $M$ can be represented as the dual of different preduals, the resulting weak-star topologies on $M$ can differ. By contrast, the weak topology on a Banach dual $E'$ is $\sigma(E',E'')$ , using all elements of the bidual, and is generally finer than $\sigma(E',E)$ .

Compactness role

For a normed space $E$ , the Banach–Alaoglu theorem says that the closed unit ball of $E'$ is compact in $\sigma(E',E)$ . This conclusion is weak-star compactness, not generally norm compactness or weak compactness. The topology is therefore central to state spaces and dual Banach spaces.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990. Publisher record. Relevant: Chapter V on weak and weak-star topologies.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Publisher record. Relevant: the dual-pair treatment of weak topologies.