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

(2)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)Where:

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

(5)## 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 $X$.*