Weak and Norm convergence in Banach Space

Question

I know (and have proven) that in a Hilbert space, $\mathscr{H}$, if a sequence $z_i\overset{w}{\to}z$ and $\|z_i\|\to\|z\|$, then $\|z_i-z\|\to0$.

I'm trying to find a counterexample in a Banach Space. It seems like $\ell_\infty$ would be a natural place to look, but I'm not having much luck.

Answer 1

$\ell_\infty$ itself is maybe not the best place, since its dual is a little unwieldy. But it has subspaces with a nice dual. So let's look at

$$c_0 = \{ x \in \ell_\infty : x_n \to 0\}.$$

We know its dual is isometrically isomorphic to $\ell_1$ per

$$\varphi_y(x) = \sum_{n=0}^\infty y_0\cdot x_0,$$

and we know that the "standard unit vectors" $e^{(k)}$ given by $e^{(k)}_n = 0$ for $n\neq k$ and $e^{(k)}_k = 1$ converge weakly to $0$.

So we could look for an $x\in c_0$ with $\lVert x+e^{(k)}\rVert_\infty \to \lVert x\rVert_\infty$.