The Weak-* Topology on X*

Definition: Let $X$ be a normed linear space. The Weak-* Topology on $X^*$ denoted by $\sigma (X^*, X)$ and is the topology induced by the subset $\hat{X} = \{ \hat{x} : x \in X \} \subseteq X^{**}$ where $\hat{x} : X^* \to \mathbb{R}$ defined for each $f \in X^*$ by $\hat{x}(f) = f(x)$.

Recall that for every $x \in X$ the function $\hat{x} : X^* \to \mathbb{R}$ defined for all $f \in X^*$ by $\hat{x}(f) = f(x)$ is a bounded linear functional on $X^*$ with $\| \hat{x} \| = \| x \|$. Therefore, $\hat{X}$ is a subset of $X^{**}$ and we can consider the topology on $X^*$ induced by $\hat{X}$ which we defined to be the weak-* topology on $X^*$.

A Local Base for f

For each $f \in X^*$, a local base for $f$ in the weak topology on $X^*$ consists of sets of the form:

(1)

\begin{align} \quad N_{\epsilon, \{ \hat{x}_1, \hat{x}_2, ..., \hat{x}_n \}}(x) &= \{ g \in X^* : |\hat{x}_k(f) - \hat{x}_k(g)| < \epsilon, \: 1 \leq k \leq n \} \\ &= \{ g \in X^* : |f(x_k) - g(x_k)| < \epsilon, \: 1 \leq k \leq n \} \end{align}

where $\epsilon > 0$ and $\{ x_1, x_2, ..., x_n \} \subseteq X$ is finite.

Weak-* Convergence in X*

Definition: Let $X$ be a normed linear space. A sequence of bounded linear functionals $(f_n)$ in $X^*$ is said to Weak-* Converge to a bounded linear functional $f \in X^*$ if for all $\hat{x} \in \hat{X}$, $\lim_{n \to \infty} \hat{x} (f_n) = \hat{x}(f)$. Equivalently, $(f_n)$ weak-* converges to $f$ if $\lim_{n \to \infty} f_n(x) = f(x)$ for each $x \in X$.

Note that a sequence of bounded linear functionals $(f_n)$ converges to $f \in X^*$ if and only if $(f_n)$ converges pointwise to $f$.