# Functional analysis

functionalfunctional analyticalgebraic function theoryanalysisApplied functional analysisfunctional analystfunctional-analyticGeneral analysisinfinite dimensionalinfinite dimensional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.wikipedia
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### Norm (mathematics)

normEuclidean normseminorm
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero.

### Inner product space

inner productinner-product spaceinner products
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis.

### Unitary operator

unitaryunitary operatorsunitarity
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, unitary etc. operators between function spaces.
In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.

### Stefan Banach

BanachBanach, StefanBanacha
Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics.

### Frigyes Riesz

F. RieszRieszFrédéric Riesz
Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
Frigyes Riesz (Riesz Frigyes, ; 22 January 1880 – 28 February 1956) was a Hungarian mathematician who made fundamental contributions to functional analysis, as did his younger brother Marcel Riesz.

### Vito Volterra

VolterraVolterra, Vito
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.

### Vector space

vectorvector spacesvectors
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.

### Linear algebra

linearlinear algebraiclinear-algebraic
In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to [[Sequence space#ℓp spaces|]].
Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions.

### Function space

function spacesfunctional spacespace of functions
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, unitary etc. operators between function spaces.
Functional analysis is organized around adequate techniques to bring function spaces as topological vector spaces within reach of the ideas that would apply to normed spaces of finite dimension.

### Banach space

Banach spacesBanachBanach-space theory
Such spaces are called Banach spaces.
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.

### Mathematical analysis

analysisclassical analysisanalytic
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

### Mathematical formulation of quantum mechanics

quantum mechanicspostulates of quantum mechanicspostulate of quantum mechanics
These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics.
Many of these structures are drawn from functional analysis, a research area within pure mathematics that was influenced in part by the needs of quantum mechanics.

### Fréchet space

Fréchet spacesFréchetentirely different meaning
More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.

### Topological vector space

topological vector spaceslinear topological spacebelow
More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.
In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

### C*-algebra

C*-algebrasC* algebraB*-algebra
These lead naturally to the definition of C*-algebras and other operator algebras.
C ∗ -algebras (pronounced "C-star") are subjects of research in functional analysis, a branch of mathematics.

### Hilbert space

Hilbert spacesHilbertseparable Hilbert space
An important example is a Hilbert space, where the norm arises from an inner product.
The success of Hilbert space methods ushered in a very fruitful era for functional analysis.

### Operator algebra

operator algebrasalgebraalgebra of bounded singular integral operators
These lead naturally to the definition of C*-algebras and other operator algebras.
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings.

### Lwów School of Mathematics

groupLwow School of Mathematicsmathematicians from the Lwów School
Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
The school was renowned for its productivity and its extensive contributions to subjects such as point-set topology, set theory and functional analysis.

### Sequence space

c'' 0 sequence spacesspace of bounded sequences
Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to [[Sequence space#ℓp spaces|]].
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers.

### Topological space

topologytopological spacestopological structure
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.

### Invariant subspace problem

invariant subspace conjectureAronszajn–Smith theoremoperator without an invariant subspace
Many special cases of this invariant subspace problem have already been proven.
In the field of mathematics known as functional analysis, the invariant subspace problem is an incompletely unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself.

### Lp space

L'' ''p'' spaceL ''p'' spacesL'' ''p'' spaces
Examples of Banach spaces are L^{\,p}-spaces for any real number p\geq1.
L p spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces.

The term was first used in Hadamard's 1910 book on that subject.
In Paris Hadamard concentrated his interests on the problems of mathematical physics, in particular partial differential equations, the calculus of variations and the foundations of functional analysis.

### Hahn–Banach theorem

Hahn-Banach theoremHahn–Banach separation theorem
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field.
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.

### Open mapping theorem (functional analysis)

open mapping theoremBanach–Schauder theorem
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.