When does the concept of "normed vector space" emerge?
Who is the first mathematician to consider this setting?
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2$\begingroup$ The earliest use Google Books found for me was 1934; by 1950 it was common. $\endgroup$ – kimchi lover Dec 18 '19 at 15:00
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$\begingroup$ @kimchilover Could you please cite some reference, particularly for the rarity? So it would make a good answer. $\endgroup$ – peterh - Reinstate Monica Dec 18 '19 at 15:50
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1$\begingroup$ @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. $\endgroup$ – kimchi lover Dec 18 '19 at 16:10
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1$\begingroup$ 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". $\endgroup$ – Dave L Renfro Dec 18 '19 at 16:25
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$\begingroup$ My bad: I misread the Q to ask for the origin of the term, not the concept. $\endgroup$ – kimchi lover Dec 18 '19 at 18:50
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A good source for these types of questions is Miller's site Earliest Known Uses of Some of the Words of Mathematics. On the norms in vector spaces we find the following:
"On page 57 of his 1908 paper “Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten,” Rendiconti del Circolo Matematico di Palermo, 25 (1908), Erhard Schmidt defines for a function A(x), x = 1, 2, 3 ..., which, he says in a footnote, can be regarded as a vector in an infinite-dimensional space, a positive quantity ||A||, the Euclidean norm, which he calls the length, and calls a vector “normirt,” if its length is equal to 1.
"In 1921 Albert A. Bennett "Normalized Geometric Systems," Proc. National Acad. Sci. U.S.A. 7 p. 84: "The notion of norm or numerical value of a complex quantity, c = a + b√- 1, namely, |c| = √(a2 + b2), as it arises in algebra, has a more or less immediate generalization to more extensive matric systems.""
"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 $\sqrt[r]{\int_a^l|\varphi|^r\,dx}$."
As Dave Renfro mentioned in the comment, the theory of normed vector spaces was developed in the 1920-s by Banach, Hahn and Wiener (concrete functional spaces with a norm, without the name, were studied before that). Their use became a standard tool Banach's 1932 book Théorie des opérations linéaires (Theory of Linear Operations, 1932), which flashed out the role of completeness and systematically developed the theory of linear operators on complete (Banach) spaces. Additional impulse was provided by von Neumann's Mathematical Foundations of Quantum Mechanics published the same year, which systematically utilized Hilbert spaces and operators on them.
