Let $(X,\|\cdot\|)$ be an infinite-dimensional normed vector space. Suppose that the weak topology of $X$ is metrizable by a metric $d$.
Denote by $B^d(x,r)$ the open balls with respect to $d$; they are therefore weakly open. We have that for every $n$ the ball $B^d(0,\frac{1}{n})$ contains a non-trivial subspace. We could then argue as follows:
Choose in each $B^d(0,\frac{1}{n})$ an $x_n$ such that $\|x_n\|=n$. We have $x_n\rightharpoonup x$ but $\|x_n\|\to \infty$. Which is an absurd, because we know that the sequence $(\|x_n\|)_{n=1}^\infty$ must be bounded.
My question lies in the fact that in the Brezis book (Exercise 3.8) there is a proof which uses Baire's theorem! I don't know why they use such a complicated demonstration when there is so much simpler proof, if my proof is correct of course.
Best Answer
The proof seems correct. The assumption that $X$ is infinite dimensional is used to show that $B^d(0,1/n)$ contains a non-trivial subspace and the argument should be detailed a little more because it is not straightforward. The set $B^d(0,1/n)$ contains an open subset of the form $O:=\bigcap_{j=1}^N\{x,|f_j(x)|<\delta\}$ for some integer $N$, $f_j\in X^*$ and $\delta >0$. There is $y\neq 0$ such that $f_j(y)= 0$ for each $j\in \{1,\dots,N\}$ because $X$ has a dimension $\geqslant N+1$.