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.