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| {{for|functional analysis as used in psychology|functional analysis (psychology)}}
| | Alyson is what my husband enjoys to call me but I don't like when individuals use my complete name. It's not a common factor but what I like performing is to climb but I don't have the time lately. Distributing manufacturing is where her primary income comes from. Some time in the past she chose to live in Alaska and her mothers and fathers live nearby.<br><br>my web site - love psychics ([http://www.atvriders.tv/uprofile.php?UID=257230 your domain name]) |
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| [[Image:Drum vibration mode12.gif|thumb|right|200px|One of the possible modes of vibration of an idealized circular [[drum head]]. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.]]
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| '''Functional analysis''' is a branch of [[mathematical analysis]], the core of which is formed by the study of [[vector space]]s endowed with some kind of limit-related structure (e.g. [[Inner product space#Definition|inner product]], [[Norm (mathematics)#Definition|norm]], [[Topological space#Definition|topology]], etc.) and the [[linear transformation|linear operator]]s acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of [[function space|spaces of functions]] and the formulation of properties of transformations of functions such as the [[Fourier transform]] as transformations defining [[continuous function|continuous]], [[unitary operator|unitary]] etc. operators between function spaces. This point of view turned out to be particularly useful for the study of [[differential equations|differential]] and [[integral equations]].
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| The usage of the word ''[[functional (mathematics)|functional]]'' goes back to the [[calculus of variations]], implying a function whose argument is a function and the name was first used in [[Jacques Hadamard|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 [[Frigyes Riesz|Riesz]] and the [[Lwów School of Mathematics|group]] of [[Poland|Polish]] mathematicians around [[Stefan Banach]].
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| In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular [[Dimension (vector space)|infinite dimensional spaces]]. In contrast, [[linear algebra]] deals mostly with finite dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of [[measure (mathematics)|measure]], [[integral|integration]], and [[probability]] to infinite dimensional spaces, also known as '''infinite dimensional analysis'''.
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| ==Normed vector spaces==
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| The basic and historically first class of spaces studied in functional analysis are [[complete space|complete]] [[normed vector space]]s over the [[real number|real]] or [[complex number]]s. Such spaces are called [[Banach space]]s. 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]].
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| More generally, functional analysis includes the study of [[Fréchet space]]s and other [[topological vector space]]s not endowed with a norm.
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| An important object of study in functional analysis are the [[continuous function (topology)|continuous]] [[linear transformation|linear operators]] defined on Banach and Hilbert spaces. These lead naturally to the definition of [[C*-algebra]]s and other [[operator algebra]]s.
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| ===Hilbert spaces===
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| [[Hilbert space]]s can be completely classified: there is a unique Hilbert space [[up to]] [[isomorphism]] for every [[cardinal number|cardinality]] of the [[orthonormal basis]]. Finite-dimensional Hilbert spaces are fully understood in [[Linear Algebra|linear algebra]], and infinite-dimensional [[Separable space|separable]] Hilbert spaces are isomorphic to [[Sequence space#ℓp spaces|<math>\ell^{\,2}(\aleph_0)\,</math>]]. 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.
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| ===Banach spaces===
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| General [[Banach space]]s are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an [[orthonormal basis]].
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| Examples of Banach spaces are [[Lp space|<math>L^{\,p}</math>-spaces]] for any real number <math>p\geq1</math> . Given also a measure <math>\mu</math> on set <math>X</math>, then <math>L^{\,p}(X)</math>, sometimes also denoted <math>L^{\,p}(X,\mu)</math> or <math>L^{\,p}(\mu)</math>, has as its vectors equivalence classes <math>[\,f\,]</math> of [[Lebesgue-measurable function|measurable function]]s whose [[absolute value]]'s <math>p</math>-th power has finite integral, that is, functions <math>f\,</math> for which one has
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| :<math>\int_{X}\left|f(x)\right|^p\,d\mu(x)<+\infty</math>.
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| If <math>\mu</math> is the [[counting measure]], then the integral may be replaced by a sum. That is, we require
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| :<math>\sum_{x\in X}\left|f(x)\right|^p<+\infty</math>.
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| Then it is not necessary to deal with equivalence classes, and the space is denoted <math>\ell^{\ p}(X)</math>, written more simply <math>\,\ell^{\,p\ }</math> in the case when <math>X</math> is the set of non-negative [[integer]]s.
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| In Banach spaces, a large part of the study involves the [[Continuous dual|dual space]]: the space of all [[continuous function (topology)|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.
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| Also, the notion of [[derivative]] can be extended to arbitrary functions between Banach spaces. See, for instance, the [[Fréchet derivative]] article.
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| </big>
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| ==Major and foundational results==
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| Important results of functional analysis include:
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| *The [[uniform boundedness principle]] (also known as [[Banach–Steinhaus theorem]]) applies to sets of operators with uniform bounds.
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| *One of the [[spectral theorem]]s (there is 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]].
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| *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.
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| *The [[open mapping theorem (functional analysis)|open mapping theorem]] and [[closed graph theorem]].
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| ''See also'': [[List of functional analysis topics]].
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| ==Foundations of mathematics considerations==
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| 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.
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| ==Points of view==
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| Functional analysis in its {{As of|2004|alt=present form}} includes the following tendencies:
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| *''Abstract analysis''. An approach to analysis based on [[topological group]]s, [[topological ring]]s, and [[topological vector space]]s.
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| *''Geometry of [[Banach space]]s'' 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.
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| *''[[Noncommutative geometry]]''. Developed by [[Alain Connes]], partly building on earlier notions, such as [[George Mackey]]'s approach to [[ergodic theory]].
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| *''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]].
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| ==See also==
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| * [[List of functional analysis topics]]
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| * [[Spectral theory]]
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| ==References==
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| * Aliprantis, C.D., Border, K.C.: ''Infinite Dimensional Analysis: A Hitchhiker's Guide'', 3rd ed., Springer 2007, ISBN 978-3-540-32696-0. Online {{doi|10.1007/3-540-29587-9}} (by subscription)
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| * Bachman, G., Narici, L.: ''Functional analysis'', Academic Press, 1966. (reprint Dover Publications)
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| * [[Stefan Banach|Banach S.]] [http://www.ebook3000.com/Theory-of-Linear-Operations--Volume-38--North-Holland-Mathematical-Library--by-S--Banach_134628.html ''Theory of Linear Operations'']. Volume 38, North-Holland Mathematical Library, 1987, ISBN 0-444-70184-2
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| * [[Haïm Brezis|Brezis, H.]]: ''Analyse Fonctionnelle'', Dunod ISBN 978-2-10-004314-9 or ISBN 978-2-10-049336-4
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| * [[John B. Conway|Conway, J. B.]]: ''A Course in Functional Analysis'', 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5
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| * [[Nelson Dunford|Dunford, N.]] and [[Jacob T. Schwartz|Schwartz, J.T.]]: ''Linear Operators, General Theory'', and other 3 volumes, includes visualization charts
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| * Edwards, R. E.: ''Functional Analysis, Theory and Applications'', Hold, Rinehart and Winston, 1965.
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| * Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: ''Functional Analysis: An Introduction'', American Mathematical Society, 2004.
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| * [[Avner Friedman|Freidman, A.]]: ''Foundations of Modern Analysis'', Dover Publications, Paperback Edition, July 21, 2010
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| * Giles,J.R.: ''Introduction to the Analysis of Normed Linear Spaces'',Cambridge University Press,2000
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| * Hirsch F., Lacombe G. - "Elements of Functional Analysis", Springer 1999.
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| * 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
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| * Kantorovitz, S.,''Introduction to Modern Analysis'', Oxford University Press,2003,2nd ed.2006.
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| * [[Kolmogorov|Kolmogorov, A.N]] and [[Sergei Fomin|Fomin, S.V.]]: ''Elements of the Theory of Functions and Functional Analysis'', Dover Publications, 1999
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| * [[Erwin Kreyszig|Kreyszig, E.]]: ''Introductory Functional Analysis with Applications'', Wiley, 1989.
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| * [[Peter Lax|Lax, P.]]: ''Functional Analysis'', Wiley-Interscience, 2002
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| * Lebedev, L.P. and Vorovich, I.I.: ''Functional Analysis in Mechanics'', Springer-Verlag, 2002
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| * Michel, Anthony N. and Charles J. Herget: ''Applied Algebra and Functional Analysis'', Dover, 1993.
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| * Pietsch, Albrecht: ''History of Banach spaces and linear operators'', Birkhauser Boston Inc., 2007, ISBN 978-0-8176-4367-6
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| * [[Michael C. Reed|Reed, M.]], [[Barry Simon|Simon, B.]]: "Functional Analysis", Academic Press 1980.
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| * Riesz, F. and Sz.-Nagy, B.: ''Functional Analysis'', Dover Publications, 1990
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| * [[Walter Rudin|Rudin, W.]]: ''Functional Analysis'', McGraw-Hill Science, 1991
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| * Schechter, M.: ''Principles of Functional Analysis'', AMS, 2nd edition, 2001
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| * Shilov, Georgi E.: ''Elementary Functional Analysis'', Dover, 1996.
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| * [[Sobolev|Sobolev, S.L.]]: ''Applications of Functional Analysis in Mathematical Physics'', AMS, 1963
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| * [[Kōsaku Yosida|Yosida, K.]]: ''Functional Analysis'', Springer-Verlag, 6th edition, 1980
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| * Vogt, D., Meise, R.: ''Introduction to Functional Analysis'', Oxford University Press, 1997.
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| ==External links==
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| {{wikibooks|Functional Analysis}}
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| * {{springer|title=Functional analysis|id=p/f042020}}
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| * [http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html Topics in Real and Functional Analysis] by [[Gerald Teschl]], University of Vienna.
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| * [http://www.math.nyu.edu/phd_students/vilensky/Functional_Analysis.pdf Lecture Notes on Functional Analysis] by Yevgeny Vilensky, New York University.
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| * [http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis] by John Aldrich University of Southampton.
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| * [http://www.youtube.com/playlist?list=PLE1C83D79C93E2266 Lecture videos on functional analysis] by [http://www.uccs.edu/~gmorrow/ Greg Morrow] from [[University of Colorado Colorado Springs]]
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| * [http://www.coursera.org/course/functionalanalysis An Introduction to Functional Analysis] on [http://www.coursera.org/ Coursera] by John Cagnol from [[Ecole Centrale Paris]]
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| {{Mathematics-footer}}
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| {{Functional Analysis}}
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| [[Category:Functional analysis| ]]
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