A Brief History of Functional Analysis

Abstract: A Brief History of Functional Analysis

Massoud Amini

Tarbiat Modares University

Fourier in his celebrated book, The Analytic Theory of Heat (1822) discussed the first example of what is now known as the problem of (inverse) Fourier transform. About the same time, Niels Abel (1823) offered a solution to the tautochrone problem in the form of an integral equation. More generally, Liouville, in his research on 2nd order linear differential equations (1837) reduced the problem to certain integral equations.

The first rigorous treatment of the general theory of integral equations was given by Ivar Fredholm (1900- 1903). Hilbert was attracted to the new theory and published a series of five papers (1904-1906). Along these, the history of Functional Analysis (a name coined by Paul Lévy in 1922) is marked by Lebesgue thesis on integration (1902), Hilbert paper on spectral theory (1906), Fréchet thesis on metric spaces (1906), Riesz papers on classical Banach spaces (1910-1911), Banach thesis on normed spaces (1922), Hahn and Banach papers on duality (1927 and 1929, independent). These were complemented by the pioneering books of Fréchet (1928) and Banach (1932).

We give a glimpse of the development of Functional Analysis by reminding the turning points of each of the above basic steps, as well as the later developments.

Mathematics

Audience: researchers in the topic

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