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Who first defined the class of locally convex topological vector spaces?

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    $\begingroup$ According to the historical notes of Bourbaki's Topological Vector Spaces, "a general definition of locally convex spaces was given by Kolmogoroff and J. von Neumann in 1935". They do not give more precise references. $\endgroup$ – abx Dec 22 '17 at 20:55
  • $\begingroup$ M. Frechet...?! $\endgroup$ – paul garrett Dec 22 '17 at 23:34
  • $\begingroup$ I was also thinking of Frechet, although I guess he is mainly credited for the idea of abstract metric space and basically the birth of modern functional analysis. $\endgroup$ – Abdelmalek Abdesselam Dec 23 '17 at 10:17
  • $\begingroup$ Frechet introduced metric vector spaces. "Topology" was not introduced yet when Frechet worked on this. $\endgroup$ – August Cleaner Dec 23 '17 at 16:07
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Attributed to von Neumann (1935) in Dieudonné (1953, p. 496; 1981, p. 218), Köthe (1956, p. 20), Schaefer (1971, p. 37), Pietsch (2007, p. 68), Narici-Beckenstein (2011, p. 82):

The first to consider LCS was von Neumann [1935, p. 4] who called them convex spaces; the term “locally convex space” was first used by Tihonov [1935, p. 768].

(To Kolmogorov (1934) they attribute general topological vector spaces. Wehausen (1938, p. 158) has a detailed comparison of the axioms in Kolmogorov, von Neumann, and Tikhonov.)

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The reference to Kolmogorov is Studia Math, 5 (1934), 29-33. In my opinion, the most reliable source of references to classical papers is the book by Dunford-Schwartz, Linear Operators. They mention also the paper of von Neumann.

Kolmogorov writes about convexity in his note.

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