Geodesic
A Characterization of Normal Distributions in Hilbert Space
Teoriâ veroâtnostej i ee primeneniâ, Tome 2 (1957) no. 4, pp. 475-477 
Yu. V. Prokhorov; M. Fish. A Characterization of Normal Distributions in Hilbert Space. Teoriâ veroâtnostej i ee primeneniâ, Tome 2 (1957) no. 4, pp. 475-477. http://geodesic.mathdoc.fr/item/TVP_1957_2_4_a5/
@article{TVP_1957_2_4_a5,
     author = {Yu. V. Prokhorov and M. Fish},
     title = {A {Characterization} of {Normal} {Distributions} in {Hilbert} {Space}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {475--477},
     year = {1957},
     volume = {2},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1957_2_4_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Theorem 1 gives an extension of the well known result of [2] to the case of random elements in Hilbert space.