Let $X$ be a locally convex topological vector space (over $\mathbb{C}$ or $\mathbb{R}$ ), and let $X^*$ be the set of continuous linear functionals on $X$ (the continuous dual of $X$ ). If $f \in X^*$ then let $p_{f}$ denote the seminorm $p_f(x) = \lvert f(x) \rvert$ , and let $p_x(f)$ denote the seminorm $p_x(f) = \lvert f(x) \rvert$ . Obviously any normed space is a locally convex topological vector space so $X$ could be a normed space.

The weak topology on $X$ is usually denoted by $\sigma(X,X^*)$ and the weak-$*$ topology on $X^*$ is usually denoted by $\sigma(X^*,X)$ . Another common notation is $(X,wk)$ and $(X^*,wk-*)$

Topology defined on a space $Y$ by seminorms $p_\iota$ , $\iota \in I$ means that we take the sets $\{ y \in Y \mid p_\iota(y) < \epsilon \}$ for all $\iota \in I$ and $\epsilon > 0$ as a subbase for the topology (that is finite intersections of such sets form the basis).

The most striking result about weak-$*$ topology is the Alaoglu's theorem which asserts that for $X$ being a normed space, a closed ball (in the operator norm) of $X^*$ is weak-$*$ compact. There is no similar result for the weak topology on $X$ , unless $X$ is a reflexive space.

Note that $X^*$ is sometimes used for the algebraic dual of a space and $X'$ is used for the continuous dual. In functional analysis $X^*$ always means the continuous dual and hence the term weak-$*$ topology.

Bibliography

1

John B. Conway. A Course in Functional Analysis, Springer-Verlag, New York, New York, 1990.