Abstract
Not only historians of mathematics but also working analysts know how seventeenth through nineteenth century mathematicians advanced from vaguer notions to the set theoretic idea of function. The celebrated Princeton Lectures in Analysis of Elias Stein and Rami Shakarchi are shaped around that history, and review it at some length in both volumes 3 and 4. Stein and Shakarchi take the set theoretic notion as their official definition of function, but the reason they review that history twice is explicitly to contrast it with the modern forms of two other classical notions of function that they use informally. Terence Tao studied with Stein in the 1990s and emphasizes an even wider view of the function concept. All these authors show both how and why these generalizations of the set theoretic notion of function suit the purposes of classical and current analysis.
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Notes
- 1.
- 2.
Key steps in the history of distributions: Leray (1934) developed a version to solve the 3 dimensional Navier-Stokes equation (Lemarié-Rieusset 2015, Ch. 12). And in 1944 Laurent Schwartz gave them a rigorous foundation using topological vector spaces. Barany (2018, p. 263) documents how early adopters of Schwartz’s distributions took them variously as “a banal trick for applied calculations, a difficult intervention in the recent theory of topological vector spaces, a profound realignment of established methods, a radical departure from familiar concepts, and many things in between.” Cartier (2021) gives a Bourbaki insider’s perspective: “this was the great talent of Schwartz: to give a simple idea that works.”
- 3.
For our purposes think of either the improper Riemann or the Lebesgue integral.
- 4.
For the history of the vibrating string see Gray (2021, Ch. 3) and many other references.
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McLarty, C. (2023). Current and Classical Notions of Function in Real Analysis. In: Chemla, K., Ferreirós, J., Ji, L., Scholz, E., Wang, C. (eds) The Richness of the History of Mathematics. Archimedes, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-031-40855-7_9
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