Attributed to von Neumann in Schaefer (1971, p. 37), Dieudonné (1981, p. 218), 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].

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.)

Attributed to von Neumann in both Schaefer (1971, p. 37) and 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].

Attributed to von Neumann in Schaefer (1971, p. 37), Dieudonné (1981, p. 218), 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].