Geodesic
On a~probabilistic characterization of the Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 421-422

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Let, in a separable Banach space $E$, a countably-Hilbert topology can be introduced so that any continuous, with respect to this topology, generalized process, is extendable to a measure in $E'$. Then it is shown that the topology in $E$ is equivalent to a pre-Hilbert one. This result is also generalized to Fréchet spaces. 
@article{TVP_1976_21_2_a21,
     author = {D. H. Mu\v{s}tari},
     title = {On a~probabilistic characterization of the {Hilbert} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {421--422},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_2_a21/}
}
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D. H. Muštari. On a~probabilistic characterization of the Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 421-422. http://geodesic.mathdoc.fr/item/TVP_1976_21_2_a21/