[Math] a normed vector space is normed closed iff it is weakly closed.

Question

The claim is

A subspace of a normed vector space is normed closed iff it is weakly closed.

I can show one direction. Strong convergence implies weak convergence, so it is weakly closed. But I have no idea about the other direction.

Thanks in advance!

Answer 1

1. "so it is weakly closed." On what assumption? Do you mean to say, "If it is norm closed, then it is weakly closed?"
2. If that is what you mean, then you must check your reasoning. If A-convergence implies B-convergence (of a net, say), then B-closed implies A-closed, not conversely.
3. "But the link only proved one direction." That is the difficult direction.