Functional analysis is a branch of

continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...

linear functionals defined on every

Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

s is

continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...

.

Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

s'' contains many topics. One is

Volume 38, North-Holland Mathematical Library, 1987, * Brezis, H.: ''Analyse Fonctionnelle'', Dunod or * Conway, J. B.: ''A Course in Functional Analysis'', 2nd edition, Springer-Verlag, 1994, * Dunford, N. and Schwartz, J.T.: ''Linear Operators, General Theory, John Wiley & Sons'', 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. * Friedman, 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, * 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'', Birkhäuser Boston Inc., 2007, * 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 * Saxe, Karen: ''Beginning Functional Analysis'', Springer, 2001 * 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 * Vogt, D., Meise, R.: ''Introduction to Functional Analysis'', Oxford University Press, 1997. * Yosida, K.: ''Functional Analysis'', Springer-Verlag, 6th edition, 1980

Topics in Real and Functional Analysis

by Gerald Teschl, University of Vienna.

Lecture Notes on Functional Analysis

by Yevgeny Vilensky, New York University.

Lecture videos on functional analysis

b

Greg Morrow

from

mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied in ...

, the core of which is formed by the study of vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...

s endowed with some kind of limit-related structure (e.g. inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

, norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

, topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, etc.) and the linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...

s defined on 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...

'' as a noun goes back to the calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra
Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis.
Biography
Born in A ...

. 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
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...

.
In modern introductory texts on 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, and does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...

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 arecomplete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...

normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...

s over the real or complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. Such spaces are called Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

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
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which ...

, machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machin ...

, partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...

, and Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...

.
More generally, functional analysis includes the study of Fréchet spaces and other topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...

s not endowed with a norm.
An important object of study in functional analysis are the continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...

linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...

s and other operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study o ...

s.
Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert spaceup to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...

isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

for every cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

of the orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...

. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to $\backslash ell^(\backslash aleph\_0)\backslash ,$. 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

GeneralBanach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

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
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...

.
Examples of Banach spaces are $L^p$-spaces for any real number Given also a measure $\backslash mu$ on set then sometimes also denoted $L^p(X,\backslash mu)$ or has as its vectors equivalence classes $;\; href="/html/ALL/s/,f\backslash ,.html"\; ;"title=",f\backslash ,">,f\backslash ,$absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...

's $p$-th power has finite integral; that is, functions $f$ for which one has
:$\backslash int\_\backslash left,\; f(x)\backslash ^p\backslash ,d\backslash mu(x)\; <\; +\backslash infty.$
If $\backslash mu$ is the counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...

, then the integral may be replaced by a sum. That is, we require
:$\backslash sum\_\backslash left,\; f(x)\backslash ^p<+\backslash infty\; .$
Then it is not necessary to deal with equivalence classes, and the space is denoted written more simply $\backslash ell^p$ in the case when $X$ is the set of non-negative integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s.
In Banach spaces, a large part of the study involves the dual space: the space of all continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

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
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article.
Linear functional analysis

Major and foundational results

There are four major theorems which are sometimes called the four pillars of functional analysis: the Hahn–Banach theorem, the open mapping theorem, the closed graph theorem and the uniform boundedness principle, also known as the Banach–Steinhaus theorem. Important results of functional analysis include:Uniform boundedness principle

The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family ofcontinuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...

s (and thus bounded operators) whose domain is a Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

, pointwise boundedness is equivalent to uniform boundedness in operator norm.
The theorem was first published in 1927 by Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...

and Hugo Steinhaus but it was also proven independently by Hans Hahn.
Theorem (Uniform Boundedness Principle). Let $X$ be anormed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" .... Suppose that $F$ is a collection of continuous linear operators from $X$ to $Y$. If for all $x$ in $X$ one has :$\backslash sup\backslash nolimits\_\; \backslash ,\; T(x)\backslash ,\; \_Y\; <\; \backslash infty,$ then :$\backslash sup\backslash nolimits\_\; \backslash ,\; T\backslash ,\; \_\; <\; \backslash infty.$

Spectral theorem

There are many theorems known as thespectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful bec ...

, but one in particular has many applications in functional analysis.
Spectral theorem. Let $A$ be a bounded self-adjoint operator on a Hilbert space $H$. Then there is a measure space $(X,\backslash Sigma,\backslash mu)$ and a real-valued essentially bounded measurable function $f$ on $X$ and a unitary operator $U:H\backslash to\; L^2\_\backslash mu(X)$ such that :$U^*\; T\; U\; =\; A\; \backslash ;$ where ''T'' is the multiplication operator: :$;\; href="/html/ALL/s/\_\backslash varphi.html"\; ;"title="\; \backslash varphi">\; \backslash varphi$ and $\backslash ,\; T\backslash ,\; =\; \backslash ,\; f\backslash ,\; \_\backslash infty$.This is the beginning of the vast research area of functional analysis called

operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...

; see also the spectral measure.
There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference in the conclusion is that now $f$ may be complex-valued.
Hahn–Banach theorem

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of somevector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...

to the whole space, and it also shows that there are "enough" normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...

to make the study of the dual space "interesting".
Hahn–Banach theorem: If $p:V\backslash to\backslash mathbb$ is asublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a semin ..., and $\backslash varphi:U\backslash to\backslash mathbb$ is alinear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...on alinear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...$U\backslash subseteq\; V$ which is dominated by $p$ on $U$; that is, :$\backslash varphi(x)\; \backslash leq\; p(x)\backslash qquad\backslash forall\; x\; \backslash in\; U$ then there exists a linear extension $\backslash psi:V\backslash to\backslash mathbb$ of $\backslash varphi$ to the whole space $V$ which is dominated by $p$ on $V$; that is, there exists a linear functional $\backslash psi$ such that :$\backslash psi(x)=\backslash varphi(x)\backslash qquad\backslash forall\; x\backslash in\; U,$ :$\backslash psi(x)\; \backslash le\; p(x)\backslash qquad\backslash forall\; x\backslash in\; V.$

Open mapping theorem

The open mapping theorem, also known as the Banach–Schauder theorem (named afterStefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...

and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...

between surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

then it is an open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...

. More precisely,:
: Open mapping theorem. If $X$ and $Y$ are Banach spaces and $A:X\backslash to\; Y$ is a surjective continuous linear operator, then $A$ is an open map (that is, if $U$ is an open set in $X$, then $A(U)$ is open in $Y$).
The proof uses the Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that t ...

, and completeness of both $X$ and $Y$ is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...

, but is true if $X$ and $Y$ are taken to be Fréchet spaces.
Closed graph theorem

The closed graph theorem states the following: If $X$ is atopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

and $Y$ is a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Briti ...

Hausdorff space, then the graph of a linear map $T$ from $X$ to $Y$ is closed if and only if $T$ is Other 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 In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...

, is usually more relevant in functional analysis. Many very important theorems require the Hahn–Banach theorem, usually proved using the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...

, although the strictly weaker Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by consid ...

suffices. The Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that t ...

, 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 ontopological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...

s, topological rings, and topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...

s.
*''Geometry of combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

approach connected with Jean Bourgain; another is a characterization of Banach spaces in which various forms of the law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...

hold.
*''Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...

''. Developed by Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vand ...

, partly building on earlier notions, such as George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
Career
Mackey earned his bachelor of arts at Rice Unive ...

's approach to ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expre ...

.
*''Connection with quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

''. Either narrowly defined as in mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...

, or broadly interpreted by, for example, Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...

, to include most types of representation theory.
See also

* List of functional analysis topics * Spectral theoryReferences

Further reading

* Aliprantis, C.D., Border, K.C.: ''Infinite Dimensional Analysis: A Hitchhiker's Guide'', 3rd ed., Springer 2007, . Online (by subscription) * Bachman, G., Narici, L.: ''Functional analysis'', Academic Press, 1966. (reprint Dover Publications) * Banach S.br>''Theory of Linear Operations''Volume 38, North-Holland Mathematical Library, 1987, * Brezis, H.: ''Analyse Fonctionnelle'', Dunod or * Conway, J. B.: ''A Course in Functional Analysis'', 2nd edition, Springer-Verlag, 1994, * Dunford, N. and Schwartz, J.T.: ''Linear Operators, General Theory, John Wiley & Sons'', 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. * Friedman, 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, * 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'', Birkhäuser Boston Inc., 2007, * 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 * Saxe, Karen: ''Beginning Functional Analysis'', Springer, 2001 * 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 * Vogt, D., Meise, R.: ''Introduction to Functional Analysis'', Oxford University Press, 1997. * Yosida, K.: ''Functional Analysis'', Springer-Verlag, 6th edition, 1980

External links

*Topics in Real and Functional Analysis

by Gerald Teschl, University of Vienna.

Lecture Notes on Functional Analysis

by Yevgeny Vilensky, New York University.

Lecture videos on functional analysis

b

Greg Morrow

from

University of Colorado Colorado Springs
The University of Colorado Colorado Springs (UCCS) is a public research university in Colorado Springs, Colorado. It is one of four campuses that make up the University of Colorado system. As of Fall 2017, UCCS had over 12,400 undergraduate a ...

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