# Normed vector space : when and who?

When does the concept of "normed vector space" emerge?

Who is the first mathematician to consider this setting?

• The earliest use Google Books found for me was 1934; by 1950 it was common. – kimchi lover Dec 18 '19 at 15:00
• @kimchilover Could you please cite some reference, particularly for the rarity? So it would make a good answer. – peterh - Reinstate Monica Dec 18 '19 at 15:50
• @peterhsaysreinstateMonica I will, but not today. Every time I check GB I get a different answer, and GB is so whimsical about dates. There is a 1940's Trans. AMS article using it; it's in Halmos's 1947 Finite Dimensional Vector Spaces. – kimchi lover Dec 18 '19 at 16:10
• I think the general consensus is that the idea of a normed space independently arose in the early 1920s by S. Banach, H. Hahn (following up on work by E. Helly), and N. Wiener. See pp. 66-68 of The development of function spaces with particular reference to their origins in integral equation theory (also here) by Michael Bernkopf (1966). Also, google these names along with "history" and "functional analysis" and "norm". – Dave L Renfro Dec 18 '19 at 16:25
• My bad: I misread the Q to ask for the origin of the term, not the concept. – kimchi lover Dec 18 '19 at 18:50

"In 1922 S. Banach defined “la norme” for an abstract linear space in “Sur les opérations dans les ensembles abstraits et leur application aux équations integrales”, Fundamenta Mathematicae, 3, pp. 135-6. Among the examples (pp. 167-8) is $$║\varphi║$$ defined by $$\sqrt[r]{\int_a^l|\varphi|^r\,dx}$$."