Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an operator is a function which operates on (or modifies another function It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, an integral equation is an equation in which an unknown function appears under an Integral sign This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. In Mathematics, a functional is traditionally a map from a Vector space to the field underlying the vector space which is usually the Real Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach. Vito Volterra ( May 3, 1860 - October 11, 1940) was an Italian Mathematician and Physicist, best known for his Stefan Banach ( Ukrainian: Степан Степанович Банах 1892–1945 was a Polish Mathematician who worked in interwar Poland and in

Contents 1 Normed vector spaces 1.1 Hilbert spaces 1.2 Banach spaces 2 Major and foundational results 3 Foundations of mathematics considerations 4 Points of view 5 References 6 External links

Normed vector spaces

In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Such spaces are called Banach spaces. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis An important example is a Hilbert space, where the norm arises from an inner product. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm. This article deals with Fréchet spaces in functional analysis In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that These lead naturally to the definition of C*-algebras and other operator algebras. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication

Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (ℵ₀) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph-null, and its morphisms. Linear algebra is the branch of Mathematics concerned with In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper invariant subspace. In Mathematics, an invariant subspace of a Linear mapping T: V &rarr V from some Vector space Many special cases have already been proven.

Banach spaces

General Banach spaces are more complicated. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis There is no clear definition of what would constitute a base, for example.

For any real number p ≥ 1, an example of a Banach space is given by "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral" (see L^p spaces). In Mathematics, measurable functions are Well-behaved functions between measurable spaces. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphism from a space into its dual's dual. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. This is explained in the dual space article. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals

Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change See, for instance, the Fréchet derivative article. In Mathematics, the Fréchet derivative is a Derivative defined on Banach spaces Named after Maurice Fréchet, it is commonly used to formalize

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. In Mathematics, the uniform boundedness principle or Banach-Steinhaus theorem is one of the fundamental results in Functional analysis. In Mathematics, the uniform boundedness principle or Banach-Steinhaus theorem is one of the fundamental results in Functional analysis.
• One of the spectral theorems (there are indeed more than one) gives an integral formula for the normal operators on a Hilbert space. In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators In Mathematics, especially Functional analysis, a normal operator on a Hilbert space H (or more generally in a C* algebra) is a This theorem is of central importance for the mathematical formulation of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons
• The Hahn-Banach theorem extends functionals from a subspace to the full space, in a norm-preserving fashion. In Mathematics, the Hahn–Banach theorem is a central tool in Functional analysis. An implication is the non-triviality of dual spaces.
• The open mapping theorem and closed graph theorem. In Functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states that if a continuous In Mathematics, the closed graph theorem is a basic result in Functional analysis which characterizes Continuous linear operators between Banach spaces

See also: List of functional analysis topics. This is a list of Functional analysis topics, by Wikipedia page

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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which Many very important theorems require the Hahn-Banach theorem, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. In Mathematics, the Hahn–Banach theorem is a central tool in Functional analysis. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. In Mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given Abstract algebra.

Points of view

Functional analysis in its present form includes the following tendencies:

• Soft analysis. "MMIV" redirects here For the Modest Mouse album see " Baron von Bullshit Rides Again " An approach to analysis based on topological groups, topological rings, and topological vector spaces;
• Geometry of Banach spaces. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis A combinatorial approach primarily due to Jean Bourgain;
• Noncommutative geometry. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Jean Bourgain (b February 28, 1954, Ostend) is a Belgian Mathematician, noted as a prolific problem-solver Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the Developed by Alain Connes, partly building on earlier notions, such as George Mackey's approach to ergodic theory;
• Connection with quantum mechanics. Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University George Whitelaw Mackey (February 1 1916 in St Louis, Missouri – March 15 2006 in Belmont, Massachusetts) was an American Mathematician Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Either narrowly defined as in mathematical physics, or broadly interpreted by, e. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. g. Israel Gelfand, to include most types of representation theory. Israïl Moiseevich Gelfand (Израиль Моисеевич Гельфанд ישראל געלפֿאַנד (born on) is a Mathematician who has contributed substantially In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

References

• Brezis, H. : Analyse Fonctionnelle, Dunod ISBN 978-2100043149 or ISBN 978-2100493364
• Conway, John 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
• Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: Functional Analysis: An Introduction, American Mathematical Society, 2004.
• 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
• Kolmogorov, A.N and Fomin, S. Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet V. : Elements of the Theory of Functions and Functional Analysis, Dover Publications, 1999
• Kreyszig, Erwin: Introductory Functional Analysis with Applications, Wiley, 1989.
• Lax, P.: Functional Analysis, Wiley-Interscience, 2002
• Lebedev, L. Peter David Lax (born May 1, 1926 in Budapest, Hungary) is a Mathematician working in the areas of pure and applied Mathematics 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.
• 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. Sobolev (masculine and Soboleva (feminine is a popular Russian Surname and may refer to the following people Leonid Sobolev : 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.

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