Geodesic
On the weak convergence of probability measures in a Banach space
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 3, Tome 260 (1999), pp. 17-30

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Necessary and sufficient conditions for the weak convergence of Borel probability measures in a separable strictly normed Banach space with continuous modulus of convexity are derived. These conditions consist in simultanious convergence of the distribution functions of the norm and of all linear continuous functionals with respect to the sequence of distributions under consideration. 
@article{ZNSL_1999_260_a1,
     author = {A. N. Baushev},
     title = {On the weak convergence of probability measures in a {Banach} space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {17--30},
     publisher = {mathdoc},
     volume = {260},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_260_a1/}
}
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A. N. Baushev. On the weak convergence of probability measures in a Banach space. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 3, Tome 260 (1999), pp. 17-30. http://geodesic.mathdoc.fr/item/ZNSL_1999_260_a1/