In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement.

($B$ is assumed to be separable Banach space,$B'$ is its topological dual space, $\mathcal P(B)$ is the space of all Radon measures on $B$.)


'The space $\mathcal P(B)$ equipped with the weak topology is known to be a complete metric space (...). Thus, in particular, in order to check that a sequence $(\mu_n)$ in $\mathcal P(B)$ converges weakly, it suffices to show that $(\mu_n)$ is relatively compact in weak topology and all possible limits are the same. The latter can be verified along linear functionals.' (in B')

I would appreciate help in finding a poof of this statement (reference), in particular why is it enough to verify that all the limits are the same along the linear functionals rather then along all contentious and bounded functionals.

  • $\begingroup$ @Norbert Is it though? I am not looking at the book now, but if I remember correctly, $\mathcal{P}(B)$ has something to do with Radon measures---is that always a Banach space? So the weak topology is not the same as discussed in Eberlein Smulian, I don't think. On the other hand, it is mentioned in the book that $\mathcal{P}(B)$ is known to be a complete metric space under the weak topology. This gives us relative sequential compactness on relatively weakly compact sets, which should be enough to go forward. $\endgroup$ – Ben W May 31 '16 at 1:38
  • $\begingroup$ I've changed the question, hopefully it is more precise. $\mathcal P(B)$ is assumed to be a complete metric space. I still don't get why it is enough to verify that all possible limits are the same along linear functionals in B', rather then all continuous and bounded functionals. I don't see how it follows from Eberlen-Shmulian, can you please expand a little? $\endgroup$ – kcx May 31 '16 at 8:02
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    $\begingroup$ @kcx, linear functionals in B' are exactly continuous linear functionals, forget about Eberlein-Smulian $\endgroup$ – Norbert May 31 '16 at 19:30
  • $\begingroup$ @Norbert, thanks for help. I might be silly here, but there are bounded and continuous functions from $B$ to the real line which are not linear. E.g. Suppose $B$ had an inner product, then $\exp( -| \langle f, \cdot \rangle| )$ would be such a function. So what confuses me is that it is sufficient to look only the linear functionals. $\endgroup$ – kcx Jun 1 '16 at 8:09
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    $\begingroup$ weak topology on $\mathcal{P}(B)$ is induced from weak topology in B. Weak topology in B is induced by the family of seminorms $|\cdot|_f:B\to\mathbb{R}:x\mapsto |f(x)|$. $\endgroup$ – Norbert Jun 1 '16 at 18:16

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