I am trying to prove the following fact:
Let $S=\{x\in V: \|x\|=1\}$ be the unit sphere of an infinite dimensional Banach space $(V,\|.\|)$. Then prove that the weak closure of $S$, $\bar{S}^{\mathcal{T}_W}$, is $B=\{x \in V: \|x\| \leq 1\}$.
My approach: Since $B$ is a closed convex subset of $V$ containing $S$, it's weakly closed as well and thus it's obvious that $\bar{S}^{\mathcal{T}_W}$ is a subset of $B$. I am trying to prove the other direction as follows.
Let $X_{\alpha}$ be a net in $S$ and $X \in V$ such that $X_\alpha \rightarrow X$ in weak topology. In other words, $f(X_\alpha) \rightarrow f(X)$ for all $f \in V^\ast$. But I am unable to prove from this fact how every $X \in B$ satisfies this condition.
Can anyone help me in this process? Though there already exists a solution for this question on this website, I want to solve it using this procedure.
