# Weak compactness

Define a map $\varphi \colon [0,1]\to C[0,1]^*$ by $\varphi(x) = \delta_x$. Then $\varphi$ is a homeomorphism for the w*-topology. Let $K$ denote the image of $\varphi$.

I have two questions:

1) Is the set $\overline{\mbox{conv}}^{\|\cdot\|}K$ compact for the weak* topology in $C[0,1]^*$ (in other words, is it weak*-closed)?

2) Is the set $\overline{\mbox{conv}}^{w^*}K$ compact in the weak topology of $C[0,1]^*$ (that is, the weak topology implemented by $C[0,1]^{**}$)?

Thank you.

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Have you tried to apply the theorem of Banach-Alaoglu ? See in Wikipedia en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem . –  MathOverview Dec 19 '12 at 13:11
I don't think $\phi$ is a homeomorphism. It is far from being surjective. –  Hui Yu Dec 19 '12 at 13:14
He means homoemorphism onto its image. –  GEdgar Dec 19 '12 at 14:39

Idea for (1).

The weak* closed convex hull of the set of $\delta_x$ is the set of all (Borel) probability measures on $[0,1]$. So your question is: can any probability measure be approximated in norm by measures with finite support?

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