$\mathcal{F}$ convex and lower continuous $\Rightarrow$ $\mathcal{F}$ weakly lower continuous

Question

I'm having troubles with one part of a problem consisting out of several subquestions and hope some of you can help me!

Let $X$ be a Banach-space and let $\mathcal{F} : X \rightarrow (-\infty,\infty]$ a convex and lower continuous functional. I have to show, that $\mathcal{F}$ is weakly lower continuous too.

We defined these types of continuity in the following way:

$\mathcal{F}$ is called lower continuous if $u_k \rightarrow u$ in X $\Rightarrow \mathcal{F}(u) \leq \liminf_{k\rightarrow \infty}\mathcal{F}(u_k)$

$\mathcal{F}$ is called weakly lower continuous if the same holds for $u_k \rightharpoonup u$.

I guess, that it might be a good idea to use one of the last statements we got in our lecture, which states that a convex subset $C$ of a Banach space $X$ is closed in strong topology if and only if $C$ is closed in the weak topology, but I even wasn't able to prove it using this lemma.

I would be grateful, if someone could help me! 🙂

Thanks in advance!

PS: I've already looked for similar questions on stackexchange and found this one and this one but both use some different definition of lower continuous, which wasn't introduced in our lecture.
So I would appreciate if someone could help solving this task using the definitions I mentioned above, due to I don't only want to solve this problem, but also want to improve my understanding of things introduced in our lecture.

Answer 1

If a set is convex and closed it is also weakly closed. The reasoning here is that the weak topology is generated by linear functionals/halfspaces, and convex sets can be written as intersections of halfspaces.

With this in mind the proof works by considering the epigraph of $f$ (which is in fact a common approach in convex analysis). Note further that a convex function is lower semicontinuous if and only if its epigraph is closed. Since $f$ is assumed to be convex and lower semicontinuous its epigraph is convex and closed (in the regular topology) which makes it closed in the weak topology and thus weakly closed, i.e. weakly lower semicontinuous.