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.
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)
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.
- 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
- 46L: Selfadjoint operator algebras (C*-algebras, von Neumann (W*-) algebras, etc.), See also 22D25
- 46M: Methods of category theory in functional analysis, See also 18-XX
- 46N: Miscellaneous applications of functional analysis, See also 47NXX
- 46S: Other (nonclassical) types of functional analysis, See also 47SXX
- 46T: Nonlinear functional analysis (See also 47Hxx, 47Jxx, 58Cxx, 58Dxx) [new in 2000]
Browse all (old) classifications for this area at the AMS.
"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".
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).
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