Alaoglu's Theorem

Show that the functional $f$ occurring in the proof of Alaoglu's Theorem is linear.

Let $\mathcal{X}$ be a LCS and let $V$ be an open neighborhood of $0$. Show that $V^\circ$ is weak-star compact in $\mathcal{X}^*$.

If $\mathcal{X}$ is a Banach space, show that there is a compact space $X$ such that $\mathcal{X}$ is isometrically isomorphic to a closed subspace of $C(X)$.