Characterization of Continuity of Linear Functional in Weak-* Topology

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Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.

Let $X^\ast$ be the normed dual space of $X$.

Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.

Let $X^{\ast \ast}$ be the second normed dual of $X$.

Then a linear functional $\phi : \struct {X^\ast, w^\ast} \to \GF$ is continuous if and only if there exists $x \in X$ such that:

$\phi = x^\wedge$

where $x^\wedge$ is the evaluation linear transformation evaluated at $x$.

That is:

$\struct {X^\ast, w^\ast}^\ast = \iota X$

Proof

Note that the weak-$\ast$ topology $w^\ast$ is generated as an initial topology by $\set {x^\wedge : x \in X}$.

This result is then given by Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals, taking $F = \set {x^\wedge : x \in X}$.

$\blacksquare$

Sources

• 2001: Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant and Václav Zizler: Functional Analysis and Infinite-Dimensional Geometry ... (previous) ... (next): Theorem $3.16$