Continuity of the duality bracket $(x,x^*) \mapsto \langle x,x^* \rangle_{X,X^*}$

Question

Let $X$ be Banach space and let $X^*$ its dual equipped with the weak* topology.

Consider the duality bracket $(X,(X^*,\textrm{weak}^*)) \to \mathbb{C}$, $(x,x^*) \mapsto \langle x, x^*\rangle_{X,X^*}$.

Do we have a continuous bilinear map, at least on a product of bounded sets ?

I am interested by the case $X=L^1(\Omega)$.

Answer 1

The evaluation is jointly continuous on $X\times B$ if $B$ is a norm-bounded subset of $X^*$. This is Corollary 6.40. in Aliprantis & Border 2006, "Infinite Dimensional Analysis". Theorem 6.38 in the same book shows that the evaluation is never jointly continuous on all of $X\times X^*$ when $X$ is infinite dimensional.