Let $E$ be a Normed Vector Space. Let $\phi : E \rightarrow ( - \infty , + \infty] $ be a function such that $\phi$ is not equivalent to $\infty$. i.e. The set $ \{ x \in E : \phi (x) \neq \infty \}$ is not empty.
The Conjugate Function $\phi^* : E^* \rightarrow ( - \infty , + \infty]$ is defined as follows:
$$ \phi^* (f) = ^{\text{sup}} _{x \in E} \{ \{f(x) - \phi (x) \} \}$$
here, $E^*$ is the Dual Space of $E$.
My textbook claims that such $\phi^*$ is lower semicontinuous. Could someone please offer a hint for me to prove this? Thank you.
Note that, given any $x$, the map $f \mapsto f(x) - \phi(x)$ is affine and continuous. In particular, the epigraph is closed and convex (in fact, weak$^*$ closed).
The epigraph of $\phi^*$ is the intersection of all these epigraphs, which makes it (weak$^*$) closed and convex. Thus $\phi^*$ is closed and (weak$^*$) lower-semicontinuous.