Functional analysis

Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. It has its historical roots in the study of transformations such as the Fourier transform and in the study of differential and integral equations. The word 'functional' goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to Volterra.

Points of view

Functional analysis as it currently stands includes a number of directions:

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• soft analysis, the approach to mathematical analysis based generally on topological groups, topological rings and topological vector spaces;
• geometry of Banach spaces, a combinatorial approach as in the work of Jean Bourgain;
• the development by Alain Connes of noncommutative geometry, based partly on previous ideas such as George Mackey's approach to ergodic theory;
• the connection with quantum mechanics, narrowly defined in mathematical physics or broadly interpreted as by Israel Gelfand to include most types of representation theory.