Suppose $X$ is a normed space and $K$ is a subset of $X$ such that $K$ is weakly compact. Show that $K$ is norm-closed and norm-bounded.
I manage to show that $K$ is norm-bounded by using the Uniform Boundedness Theorem. However, I am not sure on how to show that $K$ is norm-closed. The following is my attempt:
Since $K$ is weakly compact, we have $x^*(K)$ is compact in the scalar field of $X$, say $F$. By the Heine-Borel Theorem, $x^*(K)$ is closed. Since $x^*$ is continuous, we have $K$ is norm-closed.
Is it correct?