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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.

## Subfields

• 46A: Topological linear spaces and related structures, For function spaces, see 46EXX
• 46B: Normed linear spaces and Banach spaces; Banach lattices, For function spaces, see 46EXX
• 46C: Inner product spaces and their generalizations, Hilbert spaces , For function spaces, see 46EXX
• 46E: Linear function spaces and their duals, see also 30H05, 32E25, 46F05; for function algebras, See 46J10
• 46F: Distributions, generalized functions, distribution spaces, For distribution theory on nonlinear spaces, See 58CXX
• 46G: Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces), see also 28-XX For nonlinear functional analysis, See 47HXX, 58-XX, especially 58CXX
• 46H: Topological algebras, normed rings and algebras, Banach algebras, For group algebras, convolution algebras and measure algebras, see 43A10, 43A20
• 46J: Commutative Banach algebras and commutative topological algebras, see also 46E25
• 46K: Topological (rings and) algebras with an involution, see also 16W10
• 46M: Methods of category theory in functional analysis, See also 18-XX
• 46T: Nonlinear functional analysis (See also 47Hxx, 47Jxx, 58Cxx, 58Dxx) [new in 2000]

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

## Textbooks, reference works, and tutorials

• Zelazko, W.: "What is known and what is not known about multiplicative linear functionals", Topological vector spaces, algebras and related areas (Hamilton, ON, 1994), 102--115; Pitman Res. Notes Math. Ser., 316; Longman Sci. Tech., Harlow, 1994. MR95k:46080
• Fillmore, Peter A.: "A user's guide to operator algebras". Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. 223 pp. ISBN 0-471-31135-9 MR97i:46094
• Semmes, Stephen: "A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller", Comm. Partial Differential Equations 19 (1994), no. 1-2, 277--319. MR94j:46038
• Aupetit, Bernard: "A primer on spectral theory". Universitext. Springer-Verlag, New York, 1991. 193 pp. ISBN 0-387-97390-7 MR92c:46001
• "Reviews on Functional Analysis, 1980-1986", AMS
• Khaleelulla, S. M., "Counterexamples in topological vector spaces", Springer-Verlag, Berlin-New York, 1982, ISBN 0-387-11565-X
• Citation for Functional analysis for the practical man
• Bourbaki, N., "Topological vector spaces. chapters 1--5", Springer-Verlag, Berlin-New York, 1987. 364 pp. ISBN3-540-13627-4 (MR88g:46002); also "Théories spectrales", Hermann, Paris 1967 166 pp.
• C* algebras: lecture notes by Jones and de la Harpe.
• Brief tutorial on Banach spaces and other linear spaces (including Fréchet, Gâteaux, and other derivatives in Banach spaces).