weak-* topologyWeakTopology2005-03-08 13:28:492005-03-09 17:16:17Definitionweak topology% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{rmk}{Remark}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.
\begin{defn}
The topology on $X$ defined by the seminorms $\{ p_f \mid f \in X^* \}$ is called the {\em weak topology} and the topology on $X^*$ defined by the seminorms $\{ p_x \mid x \in X \}$ is called the {\em weak-$*$ topology}.
\end{defn}
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 {\em weak-$*$ topology}.
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\bibitem{Conway:funcanal}
John B.\@ Conway.
{\em \PMlinkescapetext{A Course in Functional Analysis}},
Springer-Verlag, New York, New York, 1990.
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