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Functional analysis is a branch of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s endowed with some kind of limit-related structure (for example,
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, ofte ...
, norm, or
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
) 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 di ...
s defined on these spaces and suitably respecting these structures. 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 In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and
integral equations In mathematical analysis, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3 ...
. The usage of the word '' functional'' 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 Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...
, 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. 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 original ...
. 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 Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theories of measure, integration, and
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
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 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 for ...
s. Such spaces are called
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
s. An important example is a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, 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, whic ...
,
machine learning Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task ( ...
,
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
, and Fourier analysis. 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 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 contin ...
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 space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
s can be completely classified: there is a unique Hilbert space
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
for every
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of the
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
. Finite-dimensional Hilbert spaces are fully understood in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, and infinite-dimensional separable Hilbert spaces are isomorphic to \ell^(\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 space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
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 (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
. Examples of Banach spaces are L^p-spaces for any real number Given also a measure \mu on set then sometimes also denoted L^p(X,\mu) or has as its vectors equivalence classes ,f\,/math> of
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s whose
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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
's p-th power has finite integral; that is, functions f for which one has \int_\left, f(x)\^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_\left, f(x)\^p < \infty . Then it is not necessary to deal with equivalence classes, and the space is denoted written more simply \ell^p in the case when X is the set of non-negative
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. In Banach spaces, a large part of the study involves the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...
: 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 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'' me ...
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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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 In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
* the open mapping theorem * the closed graph theorem * 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 In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
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 of continuous linear operators (and thus bounded operators) whose domain is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
, 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 original ...
and Hugo Steinhaus but it was also proven independently by Hans Hahn.


Spectral theorem

There are many theorems known as the
spectral theorem In 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 because computations involvin ...
, but one in particular has many applications in functional analysis. 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 operato ...
; see also the spectral measure. There is also an analogous spectral theorem for bounded
normal operator In mathematics, especially functional analysis, a normal operator on a complex number, complex Hilbert space H is a continuous function (topology), continuous linear operator N\colon H\rightarrow H that commutator, commutes with its Hermitian adjo ...
s on Hilbert spaces. The only difference in the conclusion is that now f may be complex-valued.


Hahn–Banach theorem

The
Hahn–Banach theorem In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...
"interesting".


Open mapping theorem

The open mapping theorem, also known as the Banach–Schauder theorem (named after
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 original ...
and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
s is
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
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, The proof uses the Baire category theorem, 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, but is true if X and Y are taken to be Fréchet spaces.


Closed graph theorem


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 Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
. However, a somewhat different concept, the
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 theorems require the
Hahn–Banach theorem In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
, usually proved using the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
, although the strictly weaker
Boolean prime ideal theorem In mathematics, the Boolean prime ideal theorem states that Ideal (order theory), ideals in a Boolean algebra (structure), Boolean algebra can be extended to Ideal (order theory)#Prime ideals , prime ideals. A variation of this statement for Filte ...
suffices. The Baire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.


Points of view

Functional analysis includes the following tendencies: *''Abstract analysis''. An approach to analysis based on
topological group In mathematics, topological groups are 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 structures ...
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
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
s'' contains many topics. One is
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
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 Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
''. Either narrowly defined as in
mathematical physics Mathematical physics is the development of mathematics, 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 de ...
, or broadly interpreted by, for example, Israel Gelfand, to include most types of
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
.


See also

* List of functional analysis topics *
Spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...


References


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 {{Authority control