example of a Banach space X and a subspace Y such that Y is strongly closed but not wealky closed.

Question

I am trying to solve this exercise: find a Banach space X and a subspace Y such that Y is strongly closed but is not wealky closed.

I know Y can't be a convex subspace because strongly closed + convex implies weakly closed, but then I am out of ideas on what kind of spaces to look for.

Answer 1

Assuming that subspace means possibly non-linear subspace, the cannonical basis $(e_n)$ of $c_0$ is norm closed but not weakly closed as $e_n \overset{w}{\longrightarrow} 0$.

The fact that $(e_n)$ is norm closed is obvious. To prove that it weakly converges to zero, fix $x = (x_n) \in c_0^* = \ell^1$. Then \begin{align} x(e_n) = x_n \longrightarrow 0. \end{align}