[Search][Subject Index][MathMap][Tour][Help!]

	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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

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
• 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.

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).

Software and tables

Other web sites with this focus

• Recommended: Operator Algebra Resources maintained by N. Christopher Phillips
• Banach space preprint server at Oklahoma State
• Functional Analysis preprint server at Los Alamos
• Operator algebras at the University of Northern British Columbia
• Here are the AMS and Goettingen resource pages for area 46.

Selected topics at this site

• Some basics on Topological Vector Spaces
• Reprise of basic terms (locally convex spaces, Fréchet spaces, etc.)
• Simple illustration: how do topological vector spaces arise in basic calculus questions?
• What is alignment in normed linear spaces
• What's a Schauder basis? Hamel basis? [Robert Israel]
• Schauder bases in Banach spaces (e.g. Haar functions)
• Topology of space of functions with compact support -- Banach space?
• Two nonisomorphic Banach spaces, each isomorphic to a subspace of the other
• Characterization of nonreflexive Banach spaces by convex closed subsets
• What's a 2-norm on a vector space?
• Why L^4 is more interesting than L^3.
• L^p norms converge to L^\infty norm
• Characterization of compact subsets of L_infinity
• Parallelogram Law in a Banach space implies an inner product exists
• What does the Riesz representation theorem say?
• Kwapien's theorem: norms equivalent to Hilbert space
• Reconciling different versions of the Hahn-Banach theorem; which depend on the Axiom of Choice?
• Reflexive Banach spaces and the James space (not reflexive but close!)
• What are Sobolev spaces? (Spaces of functions)
• H=w theorem (Myers and Serrin) on density of Sobolev spaces
• Spanning a space of functions: Stone's Approximation Theorem, etc.
• How to find a class of distributions closed under products?
• What are C*-algebras?
• Literature highlights: Knot theory and Functional Analysis
• Separating distributions from smooth functions with operators
• Characterize the sine function by the magnitude of all its derivatives
• Invariant measures (cylindrical, Wiener) on infinite-dimensional sphere
• Defining "measure zero" on infinite dimensional spaces (prevalence)
• Hamel bases of R over Q cannot be 'nice' (Borel, etc.)