Definition:Weak-* Topology
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Definition
Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $X^\ast$ be the topological dual space of $X$.
For each $x \in X$, define $x^\wedge : X^\ast \to \GF$ by:
- $\map {x^\wedge} f = \map f x$
Let:
- $\sigma = \set {x^\wedge : x \in X}$
Let $w^\ast$ be the initial topology on $X^\ast$ with respect to $\sigma$.
We say that $w^\ast$ is the weak-$\ast$ topology of $X^\ast$.
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $3.14$: The weak-$\ast$ topology of a dual space