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Encyclopedia > Functional analysis

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. It has its historical roots in the study of transformations, such as the Fourier transform, and in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... Partial plot of a function f. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In mathematics, the term functional is applied to certain functions. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... Vito Volterra (May 3, 1860 - October 11, 1940) was an Italian mathematician and physicist, best known for his contributions to mathematical biology. ... Stefan Banach Stefan Banach (March 30, 1892 in KrakÃ³w, Austria-Hungary now Polandâ€“ August 31, 1945 in LwÃ³w, Soviet Union - occupied Poland), was an eminent Polish mathematician, one of the moving spirits of the LwÃ³w School of Mathematics in pre-war Poland. ...

## Contents

In the modern view, functional analysis is seen as the study of 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. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... 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 any real vector space Rn. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Fig. ... This article deals with FrÃ©chet 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. These lead naturally to the definition of C*-algebras and other operator algebras. In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... C*-algebras are an important area of research in functional analysis. ... In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. ...

### Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base. 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. One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper invariant subspace. Many special cases have already been proven. In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, an invariant subspace of a linear mapping over some vector space V is a subspace W of V such that T(W) is contained in W. An invariant subspace of T is said to be T invariant. ...

### Banach spaces

General Banach spaces are more complicated. There is no clear definition of what would constitute a base, for example. In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

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 Lp spaces). In mathematics, a Lebesgue-measurable function is a real function f : R &#8594; R such that for every real number a, the set is a Lebesgue-measurable set. ... 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 spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. 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. This is explained in the dual space article. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ...

Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article. In mathematics, a derivative is the rate of change of a quantity. ... In mathematics, the FrÃ©chet derivative is a derivative defined on Banach spaces. ...

## Major and foundational results

Important results of functional analysis include:

See also: List of functional analysis topics. In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. ... In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In functional analysis, a normal operator on a Hilbert space H is a continuous linear operator N : H &#8594; H that commutes with its hermitian adjoint N*: N N* = N* N. The main importance of this concept is that the spectral theorem applies to normal operators. ... Fig. ... In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ... In mathematics, there are two theorems with the name open mapping theorem. Functional analysis In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X &#8594; Y is a surjective continuous linear operator between Banach spaces X and... In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph. ... 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. Many very important theorems require the Hahn-Banach theorem, which relies on the axiom of choice that is strictly weaker than the Boolean prime ideal theorem. In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ... 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 number of prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. ...

## Points of view

Functional analysis in its present form includes the following tendencies: 2004 is a leap year starting on Thursday of the Gregorian calendar. ...

In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G Ã— G â†’ G and the inverse operation G â†’ G are continuous maps. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R &#8594; R, where R × R carries the product topology. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... Jean Bourgain (born Ostend, February 28, 1954), is a professor of mathematics at the Institute for Advanced Study. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ... Alain Connes (born April 1, 1947) is a French mathematician, currently Professor at the College de France (Paris, France), IHES (Bures-sur-Yvette, France) and Vanderbilt University (Nashville, Tennessee). ... George Mackey is an American mathematician, working mainly in the fields of representation theory and group actions, and related parts of functional analysis. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... Fig. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... Israel Moiseevich Gelfand (Russian: ) (born in 1913) is a prolific mathematician in the field of functional analysis, which he interprets in a broad sense as the mathematics of quantum mechanics. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... Results from FactBites:

 Functional analysis - Wikipedia, the free encyclopedia (775 words) Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces.
 Functional analysis - definition of Functional analysis in Encyclopedia (665 words) Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. The word 'functional' goes back to the calculus of variations, implying a function whose argument is a function. The notion of derivative is extended to arbitrary functions between Banach spaces; it turns out that the derivative of a function at a certain point is really a continuous linear map.
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