closed unit ball in a Banach space is closed in the weak topology

Solution 1:

At first I wanted to follow the reasoning showed in "Brezis - Functional Analysis Sobolev Spaces and Partial Differential Equations" at pages 59-60, riassumed in this identity $$B_V =\bigcap_{f \in V^*,\|f\| \leq 1} \{x \in V \mid |\langle f,x \rangle | \leq 1 \}$$ (two words about it: it relies on the characterization of open/closed set in the weak topology and property of the operatorial norm and -obviously- some corollaries of HB, existence of $f_{x_0}$ such that $f(x_0)=\|x_0\|$ and $\|f_{x_0}\|=1$) and so it would be not suitable for the question here.

But there is a more deep use of HB here. An immediate corollary of HB (adapted to these hypothesis) is

Corollary. Let $V$ a non trivial Banach Space, then $V^*$ is not trivial (it contains other elements than the zero map)

Without HB there is nothing (modulo equivalent forms of it) (as far as I know at least) that grants us the non triviality of $V^*$, in the infinite dimensional case obviously. Without assuming HB the first intersection is non empty but can't be refined more than $$\bigcap_{f \in V^*,\|f\| \leq 1} \{x \in V \mid |\langle f,x \rangle | \leq 1 \} = V$$ because we don't have (we can't write down explicitly) any functional other than the zero map.

In this (rather pathological) case, the weak topology could be the trivial one, the only open set is $V$, and so $B_v$ is not closed, and this observation should show the deep problem in not assuming HB.

Solution 2:

If $x_n \to x$ weakly then we have that $\lambda x_n \to \lambda x$ for all $\lambda \in V^*$and $|\lambda x_n| \leq \|\lambda\| \|x_n\|$. Dividing both sides by $\|\lambda\|$ gives $$ \frac{|\lambda x_n|}{\|\lambda \|} \leq \|x_n\|.$$ Taking $n \to \infty$ and substituting in $\|x\| = \sup_{\lambda \in V^*} \frac{|\lambda x|}{\|\lambda \|}$ gives $$\|x\| \leq \liminf_{n \to \infty} \|x_n\|.$$Then any limit $x$ of $x_n$ with $\|x_n \| \leq 1$ for all $n$ will necessarily have $\|x\| \leq 1$.