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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 $X$).
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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 $X$.
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| 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 $p_x(f) = \lvert f(x) \rvert$. |
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. |
Obviously any normed space is a locally convex topological vector space so $X$ could be a normed space. |
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| \begin{defn} |
\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}. |
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} |
\end{defn} |
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| The weak topology on $X$ is usually denoted by $\sigma(X,X^*)$ and the weak-$*$ |
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 on $X^*$ is usually denoted by $\sigma(X^*,X)$. Another common notation is $(X,wk)$ and $(X^*,wk-*)$ |
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| 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). |
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). |
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| 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. |
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. |
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| 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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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Conway:funcanal} |
\bibitem{Conway:funcanal} |
| John B.\@ Conway. |
John B.\@ Conway. |
| {\em \PMlinkescapetext{A Course in Functional Analysis}}, |
{\em \PMlinkescapetext{A Course in Functional Analysis}}, |
| Springer-Verlag, New York, New York, 1990. |
Springer-Verlag, New York, New York, 1990. |
| \end{thebibliography} |
\end{thebibliography} |
