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Title: Math/Analysis/Functional Analysis - Functional Analysis In the Mathematical Atlas.
Functional_Analysis_Seminar Web page of the functional analysis seminar at the University of Alberta

Funktionalanalysis Institut für Analysis, Johannes Kepler Universität Linz, Austria. Seminar, publications, staff.

Geometric_Functional_Analysis_at_Penn_State Seminar; Courses; Meetings; People ; Links.

Ghent_Research_Group_for_Numerical_Functional_Analysis_and_Mathematical_Modeling Includes focus of studies, list of staff and preprints. In Belgium.

Hilbert_C*-Modules_Home_Page Bibliographies on Hilbert C*-module theory.

Hilbert\'s_Moment_Problem A result strongly suggested by computer data but no proof has been found.


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46: Functional analysis[The <b>Mathematical</b> Atlas] [Search][Subject Index][MathMap][Tour][Help!][MathMap Icon]ABOUT:[Introduction][History][Related areas][Subfields]POINTERS:[Texts][Software][Web links][Selected topics here]

46: Functional analysis

Introduction

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

History

See e.g. Jean Dieudonné, "History of Functional Analysis", North-Holland (Amsterdam) 1981Functional Analysis in Historical Perspective, A.F. Monna, HalsteadPress, Wiley, New York, 1973 (167pp)

Applications and related fields

For manifolds modeled on topological linear spaces, See 57N20, 58BXXSome questions about topological vector spaces are best stated a bitmore generally in 54: General topology; in particular, vector spaces with a distance function, especiallynormed vector spaces or, more special yet, inner product spaces are examplesof 54E: Metric spaces.[Schematic of subareas and related areas]

Subfields

46A: Topological linear spaces and related structures, For function spaces, see 46EXX46B: Normed linear spaces and Banach spaces; Banach lattices, For function spaces, see 46EXX46C: Inner product spaces and their generalizations, Hilbert spaces , For function spaces, see 46EXX46E: Linear function spaces and their duals, see also 30H05, 32E25, 46F05; for function algebras, See 46J1046F: Distributions, generalized functions, distribution spaces, For distribution theory on nonlinear spaces, See 58CXX46G: Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces), see also 28-XX For nonlinear functional analysis, See 47HXX, 58-XX, especially 58CXX46H: Topological algebras, normed rings and algebras, Banach algebras, For group algebras, convolution algebras and measure algebras, see 43A10, 43A2046J: Commutative Banach algebras and commutative topological algebras, see also 46E2546K: Topological (rings and) algebras with an involution, see also 16W1046L: Selfadjoint operator algebras (C*-algebras, von Neumann (W*-) algebras, etc.), See also 22D2546M: Methods of category theory in functional analysis, See also 18-XX46N: Miscellaneous applications of functional analysis, See also 47NXX46S: Other (nonclassical) types of functional analysis, See also 47SXX46T: 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:46080Fillmore, 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:46094Semmes, 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:46038Aupetit, 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", AMSKhaleelulla, S. M., "Counterexamples in topological vector spaces", Springer-Verlag, Berlin-New York, 1982, ISBN 0-387-11565-XCitation for Functional analysis for the practical manBourbaki, 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 PhillipsBanach space preprint server at Oklahoma StateFunctional Analysis preprint server at Los AlamosOperator algebras at the University of Northern British ColumbiaHere are the AMS and Goettingen resource pages for area 46.

Selected topics at this site

Some basics on Topological Vector SpacesReprise 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 spacesWhat'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 otherCharacterization of nonreflexive Banach spaces by convex closed subsetsWhat's a 2-norm on a vector space?Why L^4 is more interesting than L^3.L^p norms converge to L^\infty normCharacterization of compact subsets of L_infinityParallelogram Law in a Banach space implies an inner product existsWhat does the Riesz representation theorem say?Kwapien's theorem: norms equivalent to Hilbert spaceReconciling 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 spacesSpanning 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 AnalysisSeparating distributions from smooth functions with operatorsCharacterize the sine function by the magnitude of all its derivativesInvariant measures (cylindrical, Wiener) on infinite-dimensional sphereDefining "measure zero" on infinite dimensional spaces (prevalence)Hamel bases of R over Q cannot be 'nice' (Borel, etc.)You can reach this page through http://www.math-atlas.org/welcome.htmlLast modified 2002/01/14 by Dave Rusin. Mail: rusin@math.niu.edu
 

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