Functional Analysis

Wednesday, 11 April 2012

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Lecture 1: Functional Analysis

35:10

Lecture 2: Functional Analysis - Complete metric spaces

41:30

Lecture 10: Functional Analysis - Normed spaces and Banach spaces

47:12

Math Ninja : Functional Analysis Lecture 7-1

8:45

Dual Diagnosis - Functional Analysis

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Lecture 9b: Functional Analysis - Normed spaces and Banach spaces

12:54

Lecture 16a: Functional Analysis - Linear maps

24:43

Lecture 11b: Functional Analysis

12:17

Functional Analysis of Autumn Leaves (Scot Gormley).m4v

38:34

Lecture 6a: Functional Analysis - Discussion session on partially ordered sets and vector spaces

12:23

Lecture 16b: Functional Analysis - Sequence spaces

17:45

Lecture 18b: Functional Analysis - Isomorphisms

7:08

Lecture 15a: Functional Analysis - Final discussion of Equivalence of norms

10:08

Lecture 13a: Functional Analysis - Normed spaces and Banach spaces

9:02

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Lecture 7: Functional Analysis - Infinite products and Tychonoff's theorem

48:06

Functional Analysis SF2707 Ari Laptev 5:3

40:09

Lecture 5b: Functional Analysis - Infinite products and Tychonoff's theorem

10:44

Lecture 4: Functional Analysis - Complete metric spaces continued

45:13

Lecture 13b: Functional Analysis - Equivalence of norms

31:45

Lecture 11a: Functional Analysis

26:41

Lecture 3: Functional Analysis - revision of Metric and Topological Spaces

44:33

Lecture 5a: Functional Analysis - Nowhere dense sets

32:31

Lecture 6b: Functional Analysis - Continuation of discussion session on partially ordered sets

32:40

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The usage of the word ''functional'' goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that subject. However, the general concept of functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.

In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces. In contrast, linear algebra deals mostly with finite dimensional spaces, or does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

Normed vector spaces

The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics.

More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebras and other operator algebras.

Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to

\ell^{\,2}(\aleph_0)\,

. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven.

Banach spaces

General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, Banach spaces lack a notion analogous to an orthonormal basis.

Examples of Banach spaces are $L^{\,p}$ -spaces for any real number $p\geq1$ . Given also a measure $\mu$ on set $X$ , then $L^{\,p}(X)$ , sometimes also denoted $L^{\,p}(X,\mu)$ or $L^{\,p}(\mu)$ , has as its vectors equivalence classes $[\,f\,]$ of measurable functions whose absolute value's $p$ -th power has finite integral, that is, functions $f\,$ for which one has : $\int_{X}\left|f(x)\right|^p\,d\mu(x)<+\infty$ .

If $\mu$ is the counting measure, then the integral may be replaced by a sum. That is, we require : $\sum_{x\in X}\left|f(x)\right|^p<+\infty$ .

Then it is not necessary to deal with equivalence classes, and the space is denoted $\ell^{\ p}(X)$ , written more simply $\,\ell^{\,p\ }$ in the case when $X$ is the set of non-negative integers.

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. This is explained in the dual space article.

Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article.

Major and foundational results

Important results of functional analysis include:

The uniform boundedness principle (also known as Banach–Steinhaus theorem) applies to sets of operators with uniform bounds.

One of the spectral theorems (there are indeed more than one) gives an integral formula for the normal operators on a Hilbert space. This theorem is of central importance for the mathematical formulation of quantum mechanics.

The Hahn–Banach theorem extends functionals from a subspace to the full space, in a norm-preserving fashion. An implication is the non-triviality of dual spaces.

The open mapping theorem and closed graph theorem.

''See also'': List of functional analysis topics.

Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basis for such spaces may require Zorn's lemma. However, a somewhat different concept, Schauder basis, is usually more relevant in functional analysis. Many very important theorems require the Hahn–Banach theorem, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.

Points of view

Functional analysis in its includes the following tendencies:

''Abstract analysis''. An approach to analysis based on topological groups, topological rings, and topological vector spaces;

''Geometry of Banach spaces'' contains many topics. One is combinatorial approach connected with Jean Bourgain; another is a characterization of Banach spaces in which various forms of the law of large numbers hold.

''Noncommutative geometry''. Developed by Alain Connes, partly building on earlier notions, such as George Mackey's approach to ergodic theory.

''Connection with quantum mechanics''. Either narrowly defined as in mathematical physics, or broadly interpreted by, e.g. Israel Gelfand, to include most types of representation theory.

References

Aliprantis, C.D., Border, K.C.: ''Infinite Dimensional Analysis: A Hitchhiker's Guide'', 3rd ed., Springer 2007, ISBN 978-3-540-32696-0. Online (by subscription)

Brezis, H.: ''Analyse Fonctionnelle'', Dunod ISBN 978-2-10-004314-9 or ISBN 978-2-10-049336-4

Conway, J. B.: ''A Course in Functional Analysis'', 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5

Dunford, N. and Schwartz, J.T.: ''Linear Operators, General Theory'', and other 3 volumes, includes visualization charts

Edwards, R. E.: ''Functional Analysis, Theory and Applications'', Hold, Rinehart and Winston, 1965.

Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: ''Functional Analysis: An Introduction'', American Mathematical Society, 2004.

Freidman, A.: ''Foundations of Modern Analysis'', Dover Publications, Paperback Edition, July 21, 2010

Giles,J.R.: ''Introduction to the Analysis of Normed Linear Spaces'',Cambridge University Press,2000

Hirsch F., Lacombe G. - "Elements of Functional Analysis", Springer 1999.

Hutson, V., Pym, J.S., Cloud M.J.: ''Applications of Functional Analysis and Operator Theory'', 2nd edition, Elsevier Science, 2005, ISBN 0-444-51790-1

Kantorovitz, S.,''Introduction to Modern Analysis'', Oxford University Press,2003,2nd ed.2006.

Kolmogorov, A.N and Fomin, S.V.: ''Elements of the Theory of Functions and Functional Analysis'', Dover Publications, 1999

Kreyszig, E.: ''Introductory Functional Analysis with Applications'', Wiley, 1989.

Lax, P.: ''Functional Analysis'', Wiley-Interscience, 2002

Lebedev, L.P. and Vorovich, I.I.: ''Functional Analysis in Mechanics'', Springer-Verlag, 2002

Michel, Anthony N. and Charles J. Herget: ''Applied Algebra and Functional Analysis'', Dover, 1993.

Pietsch, Albrecht: ''History of Banach spaces and linear operators'', Birkhauser Boston Inc., 2007, ISBN 978-0-8176-4367-6

Reed, M., Simon, B.: "Functional Analysis", Academic Press 1980.

Riesz, F. and Sz.-Nagy, B.: ''Functional Analysis'', Dover Publications, 1990

Rudin, W.: ''Functional Analysis'', McGraw-Hill Science, 1991

Schechter, M.: ''Principles of Functional Analysis'', AMS, 2nd edition, 2001

Shilov, Georgi E.: ''Elementary Functional Analysis'', Dover, 1996.

Sobolev, S.L.: ''Applications of Functional Analysis in Mathematical Physics'', AMS, 1963

Yosida, K.: ''Functional Analysis'', Springer-Verlag, 6th edition, 1980

External links

Functional Analysis by Gerald Teschl, University of Vienna.

Lecture Notes on Functional Analysis by Yevgeny Vilensky, New York University.

Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis by John Aldrich University of Southampton.

ar:تحليل دالي bg:Функционален анализ ca:Anàlisi funcional cs:Funkcionální analýza da:Funktionalanalyse de:Funktionalanalysis es:Análisis funcional fa:آنالیز تابعی fr:Analyse fonctionnelle (mathématiques) gl:Análise funcional ko:함수해석학 it:Analisi funzionale he:אנליזה פונקציונלית ka:ფუნქციონალური ანალიზი kk:Функционалдық талдау nl:Functionaalanalyse ja:関数解析学 pms:Anàlisi fonsional pl:Analiza funkcjonalna pt:Análise funcional ru:Функциональный анализ sq:Analiza funksionale sk:Funkcionálna analýza fi:Funktionaalianalyysi sv:Funktionalanalys tr:Fonksiyonel analiz uk:Функціональний аналіз vi:Giải tích hàm zh:泛函分析

This text is licensed under the Creative Commons CC-BY-SA License. This text was originally published on Wikipedia and was developed by the Wikipedia community.

name	Ari Laptev
birth date	August 10, 1950
birth place	Kiev, Ukraine
nationality
citizenship	Sweden
fields	Mathematics
workplaces	Royal Institute of Technology
alma mater	St. Petersburg State University
doctoral advisor	Michael Solomyak
doctoral students	Rupert Frank Dimitri Gioev Jesper Tidblom
awards	Royal Society Wolfson Research Merit Award (2007) }}

48:06

Lecture 7: Functional Analysis - Infinite products and Tychonoff's theorem

NottmUniversity

40:09

Functional Analysis SF2707 Ari Laptev 5:3

galosstiftelsen

10:44

Lecture 5b: Functional Analysis - Infinite products and Tychonoff's theorem

NottmUniversity

45:13

Lecture 4: Functional Analysis - Complete metric spaces continued

NottmUniversity

31:45

Lecture 13b: Functional Analysis - Equivalence of norms

NottmUniversity

26:41

Lecture 11a: Functional Analysis

NottmUniversity

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Normed vector spaces

Hilbert spaces

Banach spaces

Major and foundational results

Foundations of mathematics considerations

Points of view

See also

References

External links

References

Other sources

1. Personal Information Collection and Use

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3. Third Party Advertisers

4. Business Transfers