A Brief History of Functional Analysis Link246;ping （泛函分析简史链接246;平）.pdfVIP

A Brief History of Functional Analysis Functional analysis was born in the early years of the twentieth century as part of a larger trend toward abstraction—what some authors have called the “arithmetization” of analysis. This same trend toward “axiomatics” contributed to the foundations of abstract linear algebra, modern geometry, and topology. Functional analysis is now a very broad field, encompassing much of modern analysis. In fact, it would be difficult to give a simple definition of what functional analysis means today. Rather than discuss its current meaning, we will concentrate on its foundations and settle for an all too brief description of modern trends. We will discuss several episodes from the early history of “abstract analysis,” especially those related to the development of vector spaces and other “abstract spaces.” Of particular interest to us will be the movement from the specific to the generic in mathematics; as one example of this, you may be surprised to learn that the practice of referring to functions by “name,” writing a single letter f, say, rather than the referring to its values f (x), only became common in our own century. In particular, we will discuss the work of Fredholm and Hilbert on integral equations and operator theory, the work of Volterra and Hadamard on the problem of moments, the work of Lebesgue, Fr´echet, and Riesz on abstract spaces, and the work of Helly, Hahn, and Banach on the notion of duality. In addition, we will present a few examples which illustrate the functional analytic viewpoint. 1. Brief Summary of Important Dates — Fredholm’s 1900 paper on integral equations — Lebesgue’s 1902 thesis on integration — Hilbert’s paper of 1906 on spectral theory — Fr´echet’s 1906 thesis on metric spaces — Riesz’s papers of 1910 and 1911 on C [a, b ] and Lp — Helly’s papers of 1912 and 1921 — Banach’