Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
Have you tried to apply the theorem of Banach-Alaoglu ? See in Wikipedia en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem . –  Elias 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
add comment

1 Answer

up vote 1 down vote accepted

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?

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.