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46: Functional analysis


Introduction

Functional analysis views the big picture in differential equations, for example, thinking of a differential operator as a linear map on a large set of functions. Thus this area becomes the study of (infinite-dimensional) vector spaces with some kind of metric or other structure, including ring structures (Banach algebras and C-* algebras for example). Appropriate generalizations of measure, derivatives, and duality also belong to this area.

History

See e.g. Jean Dieudonné, "History of Functional Analysis", North-Holland (Amsterdam) 1981

Functional Analysis in Historical Perspective, A.F. Monna, Halstead Press, Wiley, New York, 1973 (167pp)

Applications and related fields

For manifolds modeled on topological linear spaces, See 57N20, 58BXX

Some questions about topological vector spaces are best stated a bit more generally in 54: General topology; in particular, vector spaces with a distance function, especially normed vector spaces or, more special yet, inner product spaces are examples of 54E: Metric spaces. [Schematic of subareas and related areas]

Subfields

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

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