weak-* topology

Let $X$ be a locally convex topological vector space (over $ℂ$ or $ℝ$), 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 $p_f(x)=|f(x)|$, and let $p_x(f)$ denote the seminorm $p_x(f)=|f(x)|$.Obviously any normed space is a locally convex topological vector space so $X$ could be a normed space.

Definition.

The topology on $X$ defined by the seminorms $\{p_f\mid f\in X^{*}\}$ is called the weak topology and the topology on $X^{*}$ defined by the seminorms $\{p_x\mid x\in X\}$ is called the weak-$\mathrm{*}$ topology.

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 for all $\iota \in I$ and $ϵ>0$ 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 $X^{\prime }$ is used for the continuous dual. In functional analysis $X^{*}$ always means the continuous dual and hence the term weak-$\mathrm{*}$ topology.