A Comparison of the Weak and Weak* Topologies

# A Comparison of the Weak and Weak* Topologies

We will now compare the weak and weak* topologies.

Topology Alternative Names Defined on Description
The Weak Topology The $X^*$-Weak Topology $X$ The weakest topology which makes all the linear functionals in $X^*$ continuous with respect to the topology.
The Weak* Topology The $J(X)$-Weak Topology $X^*$ The weakest topology which makes all the linear functionals in $J(X)$ continuous with respect to the topology. Each linear functional in $J(X)$ is of the form $J_x : X^* \to \mathbb{C}$ defined for all $\varphi \in X^*$ by $J_x(\varphi) = \varphi(x)$.

Observe that weak can also define the weak topology on $X*$. In that case, the weak topology on $X*$ is the weaest topology which makes all linear functionals in $X^{**}$ continuous with respect to the topology. Since every linear functional in $J(X)$ is already continuous with respect to this topology, we see that:

(1)
\begin{align} \quad \mathrm{weak* \: topology \: on \:} X^* \subseteq \mathrm{weak \: topology \: on \:} X \end{align}