Let $X$ be a Banach space. === Fact 1 Let $S\subset X$ be non-empty closed subset. If $\{x_n\}\subset S$ and $x_n$ converges to $x$ weakly. Then $x\in S$. //Proof:// Suppose $x\notin S$, then $\text{dict}(x, S)>0$. Using this and there exists a affine function strictly separate $x$ and $S$, which contradict with that $S\supset x_n \rightharpoonup x$. === Fact 2 Let $F$ be a convex and lower-semi-continuous function (norm topology) on $X$, then $F$ is also lower semi-continuous with respect to the weak topology. //Proof:// The proof directly follows from the Mazur's lemma. === Mazur's lemma It says that any weakly convergent sequence in a Banach space has a sequence of convex combinations of its members that converges strongly to the same limit. see [[http://en.wikipedia.org/wiki/Mazur's_lemma|Mazur's lemma - Wikipedia]]