The Weak Topologies on X and X*

The Weak Topologies on X and X*

The Weak Topology on X

Recall from The Weak Topology Induced by W ⊆ X♯ page that if $X$ is a linear space then we could take any subset $W \subseteq X^{\sharp}$ of linear functionals on $X$ and consider the weak topology induced by $W$. We proved that the only linear functionals that were $W$-weakly continuous were functionals in $W$ itself.

If $X$ is more than just a linear space, i.e., if $X$ is a normed linear space then we can let $W = X^*$ - the set of all bounded linear functionals. The weak topology induced by $X^*$ is given a special name which we formally define below.

 Definition: Let $X$ be a normed linear space. The Weak Topology on $X$ denoted by $\sigma (X, X^*)$ is defined to be the weak topology on $X$ induced by $X^*$.

By the proposition referenced above, we note that only linear functionals that are weakly continuous are bounded linear functionals. Thus, $X$ with the weak topology is the weakest topology which makes every bounded linear functional weakly continuous. Moreover, note that the weak topology is weaker than the topology induced by the norm on $X$ (hence its name).

Local Base for x

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

(1)
\begin{align} \quad N_{\epsilon, \{ f_1, f_2, ..., f_n \}}(x) = \{ y \in X : |f_k(x) - f_k(y)| < \epsilon, \: 1 \leq k \leq n \} \end{align}

where $\epsilon > 0$ and $\{ f_1, f_2, ..., f_n \} \subset X^*$ is finite.

Weak Convergence in X

As mentioned several times before, it is often convenient to describe exactly what it means for a sequence of points $(x_n)$ to weakly converge to $x \in X$.

 Definition: Let $X$ be a normed linear space. A sequence of points $(x_n)$ in $X$ Weakly Converges to a point $x \in X$ if for all $f \in X^*$ we have that $\displaystyle{\lim_{n \to \infty} f(x_n) = f(x)}$.

The Weak Topology on X*

Let $X$ be a normed linear space. Then $X^*$ is well-defined. We can consider the collection $X^{**}$ of bounded linear functionals on $X^*$, and the topology on $X^*$ induced by $X^{**}$.

 Definition: Let $X$ be a normed linear space. The Weak Topology on $X^*$ denoted by $\sigma (X^*, X^{**})$ is defined to be the weak topology on $X^*$ induced by $X^{**}$.

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:

(2)
\begin{align} \quad N_{\epsilon, \{ \varphi_1, \varphi_2, ..., \varphi_n \}}(x) = \{ g \in X^* : |\varphi_k(f) - \varphi_k(g)| < \epsilon, \: 1 \leq k \leq n \} \end{align}

where $\epsilon > 0$ and $\{ \varphi_1, \varphi_2, ..., \varphi_n \} \subset X^{**}$ is finite.

Weak Convergence in X*

We can also characterize what it means for a sequence of bounded linear functionals $(f_n)$ to weakly convergent to $f \in X^*$.

 Definition: Let $X$ be a normed linear space. A sequence of bounded linear functionals $(f_n)$ in $X^*$ Weakly Converges to a bounded linear functional $f \in X^*$ if for all $\varphi \in X^{**}$ we have that $\displaystyle{\lim_{n \to \infty} \varphi(f_n) = \varphi(f)}$.