$ f(x^*)=\sup_{x\in C}{\langle x^*,x\rangle} $ is $\sigma(X^*,X)$-lower semicontinuous.

Question

Let $X$ be a Banach space and $C\subset X$ be a nonempty, closed, and convex set.

Let $f:X^{*}\to \mathbb{R}\cup\{+\infty\}$ defined by: $$ f(x^*)=\sup_{x\in C}{\langle x^*,x\rangle} $$

Show that : $f$ is $\sigma(X^*,X)$-lower semicontinuous.

An idea please.

Answer 1

for any $a\in \mathbb R$, \begin{align} \{x^*: f(x^*)\leq a\}&=\{x^*:\sup_{x\in C}\langle x, x^*\rangle\leq a\}\\ &=\bigcap_{x\in C}\{x^*:\langle x, x^*\rangle\leq a\} \end{align}