Abstract harmonic analysis has evolved over the past few decades on the basis of several theories. First we have the classical theory of Fourier series and integrals, set forth in many treatises, such as Zygmund  and Bochner . Second, we have the algebraic theory of groups and their representations, which is also described in many standard texts (e.g. van der Waerden ). Third, we have the theory of topological spaces, by now a fundamental tool of analysis, and also the subject of standard texts (e.g. Kelley ). The latter two subjects were combined to form the notion of a topological group. This is an entity which is both a group and a topological space and in which the group operations and the topology are appropriately connected. The structure of topological groups was extensively studied in the years 1925–1940; and the subject is far from dead even today
KeywordsTopological Space Normal Subgroup Closed Subset Topological Group Cardinal Number
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