The Weak* (J(X)-Weak) Topo. on the Topo. Dual of a Norm.Lin. Sp.

The Weak* (J(X)-Weak) Topology on the Topological Dual of a Normed Linear Space

Recall that on The Second Topological Dual of a Normed Linear Space page that if $X$ is a normed linear space then the second topological dual denoted $X^{**}$ is the topological dual of the topological dual $X^*$.

Also recall that we defined a map $J : X \to X^{**}$ defined for all $x \in X$ by:

(1)
\begin{align} \quad J(x) = J_x \end{align}

Where $J_x : X^* \to \mathbb{C}$ is defined for all continuous linear functionals $\varphi \in X^*$ by:

(2)
\begin{align} \quad J_x(\varphi) = \varphi(x) \end{align}

We now define the weak* topology on the topological complement $X^*$.

Definition: Let $X$ be a normed linear space. Then the Weak* Topology on the topological complement is the $J(X)$-weak topology.

A base for the weak* topology on $X^*$ is given by:

(3)
\begin{align} \quad \mathcal B = \{ V_{\epsilon, F, \varphi} : \epsilon > 0, F \subseteq X \: \mathrm{is \: finite}, \varphi \in X^* \} \end{align}

Where:

(4)
\begin{align} \quad V_{\epsilon, F, \varphi} &= \{ \psi \in X^* : |(J(X))(\varphi) - (J(X))(\psi)| < \epsilon, \: \forall x \in F \} \\ &= \{ \psi \in X^* : |\varphi(x) - \psi(x)| < \epsilon, \: \forall x \in F \} \end{align}

For each $\varphi \in X$ a local base of $x$ is given by:

(5)
\begin{align} \quad \mathcal B_x = \{ V_{\epsilon, F, \varphi} : \epsilon > 0, F \subseteq X \: \mathrm{is \: finite} \} \end{align}

Weak* Convergence of Sequences of Continuous Linear Functionals

Definition: Let $X$ be a normed linear space. A sequence $(\varphi_n)_{n=1}^{\infty}$ in $X^*$ is said to Weak* Converge to $\varphi \in X^*$ if $(\varphi_n(x))_{n=1}^{\infty}$ converges to $\psi(x)$ for every $x \in X$.

From above, we see that a sequence of linear functionals in $X^*$ weak* converges to a linear functional $\psi$ if and only if that sequence of linear functionals converges pointwise on all of $X$.

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