# weak-* topology

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.

###### 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-$*$ 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 $\{y\in Y\mid p_{\iota}(y)<\epsilon\}$ 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 $X^{\prime}$ is used for the continuous dual. In functional analysis $X^{*}$ always means the continuous dual and hence the term weak-$*$ topology.

## References

• 1 John B. Conway. , Springer-Verlag, New York, New York, 1990.
Title weak-* topology WeakTopology 2013-03-22 15:07:05 2013-03-22 15:07:05 jirka (4157) jirka (4157) 8 jirka (4157) Definition msc 46A03 weak-* topology weak-$*$ topology weak-star topology WeakHomotopyAdditionLemma weak topology