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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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    He means homoemorphism onto its image.2012-12-19

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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?