Does a function $f: E \rightarrow \mathbb R$ and such that $f$ is continuous, but not weakly lower semcontinuous exist?

Question

Do an infinite Banach space $E$ and a function $f: E \rightarrow \mathbb R$ such that $f$ is continuous, but not weakly lower semcontinuous exist?

I'm trying to find an example, but I can find only continuous functions that are weakly lower semcontinuous.

Answer 1

Let $H$ be a Hilbert sapce wit an infinite orthonormal set $\{e_1,e_2,..\}$. Let $f(x)=-\|x\|$. Then $f$ is continuous but not weakly lower semi-continuous. Look $\{x: f(x) >-\frac 1 2\}$ to prove this. [ $0$ is in this set but none of the $e_n$'s are. Of course, $e_n \to 0$ weakly].