Glossary of functional analysis
# Glossary of functional analysis

This is a glossary for the terminology in a mathematical field of functional analysis.

Throughout the article, unless stated otherwise, the base field of a vector space is the field of real numbers or that of complex numbers. Algebras are not assumed to be unital.

* A
* B
* C
* D
* E
* F
* G
* H
* I
* J
* K
* L
* M
* N
* O
* P
* Q
* R
* S
* T
* U
* V
* W
* XYZ
* See also
* References

See also: List of Banach spaces.

    *-homomorphism between involutive Banach algebras is an algebra homomorphism preserving *.

## A

### abelian

    Synonymous with "commutative"; e.g., an abelian Banach algebra means a commutative Banach algebra.

### Alaoglu

    Alaoglu's theorem states that the closed unit ball in a normed space is compact in the weak-* topology.

### adjoint

    The adjoint of a bounded linear operator $T:H_{1}\to H_{2}$ between Hilbert spaces is the bounded linear operator $T^{*}:H_{2}\to H_{1}$ such that $\langle Tx,y\rangle =\langle x,T^{*}y\rangle$ for each $x\in H_{1},y\in H_{2}$.

### approximate identity

    In a not-necessarily-unital Banach algebra, an approximate identity is a sequence or a net $\{u_{i}\}$ of elements such that $u_{i}x\to x,xu_{i}\to x$ as $i\to \infty$ for each x in the algebra.

### approximation property

    A Banach space is said to have the approximation property if every compact operator is a limit of finite-rank operators.

## B

### Baire

    The Baire category theorem states that a complete metric space is a Baire space; if $U_{i}$ is a sequence of open dense subsets, then $\cap _{1}^{\infty }U_{i}$ is dense.

### Banach

    1. A Banach space is a normed vector space that is complete as a metric space.
    2. A Banach algebra is a Banach space that has a structure of a possibly non-unital associative algebra such that
       $\|xy\|\leq \|x\|\|y\|$ for every $x,y$ in the algebra.

### Bessel

    Bessel's inequality states: given an orthonormal set S and a vector x in a Hilbert space, where the equality holds if and only if S is an orthonormal basis; i.e., maximal orthonormal set.
       $\sum _{u\in S}|\langle x,u\rangle |^{2}\leq \|x\|^{2}$,

### bounded

    A bounded operator is a linear operator between Banach spaces for which the image of the unit ball is bounded.

### Birkhoff orthogonality

    Two vectors x and y in a normed linear space are said to be Birkhoff orthogonal if $\|x+\lambda y\|\geq \|x\|$ for all scalars λ. If the normed linear space is a Hilbert space, then it is equivalent to the usual orthogonality.

## C

### Calkin

    The Calkin algebra on a Hilbert space is the quotient of the algebra of all bounded operators on the Hilbert space by the ideal generated by compact operators.

### Cauchy–Schwarz inequality

    The Cauchy–Schwarz inequality states: for each pair of vectors $x,y$ in an inner-product space,
       $|\langle x,y\rangle |\leq \|x\|\|y\|$.

### closed

    The closed graph theorem states that a linear operator between Banach spaces is continuous (bounded) if and only if it has closed graph.

### commutant

    1. Another name for "centralizer"; i.e., the commutant of a subset S of an algebra is the algebra of the elements commuting with each element of S and is denoted by $S'$.
    2. The von Neumann double commutant theorem states that a nondegenerate *-algebra ${\mathfrak {M}}$ of operators on a Hilbert space is a von Neumann algebra if and only if ${\mathfrak {M}}''={\mathfrak {M}}$.

### compact

    A compact operator is a linear operator between Banach spaces for which the image of the unit ball is precompact.

### C*

    A C* algebra is an involutive Banach algebra satisfying $\|x^{*}x\|=\|x^{*}\|\|x\|$.

### convex

    A locally convex space is a topological vector space whose topology is generated by convex subsets.

### cyclic

    Given a representation $(\pi ,V)$ of a Banach algebra $A$, a cyclic vector is a vector $v\in V$ such that $\pi (A)v$ is dense in $V$.

## D

### direct

    Philosophically, a direct integral is a continuous analog of a direct sum.

## F

### factor

    A factor is a von Neumann algebra with trivial center.

### faithful

    A linear functional $\omega$ on an involutive algebra is faithful if $\omega (x^{*}x)\neq 0$ for each nonzero element $x$ in the algebra.

### Fréchet

    A Fréchet space is a topological vector space whose topology is given by a countable family of seminorms (which makes it a metric space) and that is complete as a metric space.

### Fredholm

    A Fredholm operator is a bounded operator such that it has closed range and the kernels of the operator and the adjoint have finite-dimension.

## G

### Gelfand

    1. The Gelfand–Mazur theorem states that a Banach algebra that is a division ring is the field of complex numbers.
    2. The Gelfand representation of a commutative Banach algebra $A$ with spectrum $\Omega (A)$ is the algebra homomorphism $F:A\to C_{0}(\Omega (A))$, where $C_{0}(X)$ denotes the algebra of continuous functions on $X$ vanishing at infinity, that is given by $F(x)(\omega )=\omega (x)$. It is a *-preserving isometric isomorphism if $A$ is a commutative C*-algebra.

### Grothendieck

    Grothendieck's inequality.

## H

### Hahn–Banach

    The Hahn–Banach theorem states: given a linear functional $\ell$ on a subspace of a complex vector space V, if the absolute value of $\ell$ is bounded above by a seminorm on V, then it extends to a linear functional on V still bounded by the seminorm. Geometrically, it is a generalization of the hyperplane separation theorem.

### Hilbert

    1. A Hilbert space is an inner product space that is complete as a metric space.
    2. In the Tomita–Takesaki theory, a (left or right) Hilbert algebra is a certain algebra with an involution.

### Hilbert–Schmidt

    1. The Hilbert–Schmidt norm of a bounded operator $T$ on a Hilbert space is $\sum _{i}\|Te_{i}\|^{2}$ where $\{e_{i}\}$ is an orthonormal basis of the Hilbert space.
    2. A Hilbert–Schmidt operator is a bounded operator with finite Hilbert–Schmidt norm.

## I

### index

    1. The index of a Fredholm operator $T:H_{1}\to H_{2}$ is the integer $\operatorname {dim} (\operatorname {ker} (T^{*}))-\operatorname {dim} (\operatorname {ker} (T))$.
    2. The Atiyah–Singer index theorem.

### index group

    The index group of a unital Banach algebra is the quotient group $G(A)/G_{0}(A)$ where $G(A)$ is the unit group of A and $G_{0}(A)$ the identity component of the group.

### inner product

    1. An inner product on a real or complex vector space $V$ is a function $\langle \cdot ,\cdot \rangle :V\times V\to \mathbb {R}$ such that for each $v,w\in V$, (1) $x\mapsto \langle x,v\rangle$ is linear and (2) $\langle v,w\rangle ={\overline {\langle w,v\rangle }}$ where the bar means complex conjugate.
    2. An inner product space is a vector space equipped with an inner product.

### involution

    1. An involution of a Banach algebra A is an isometric endomorphism $A\to A,\,x\mapsto x^{*}$ that is conjugate-linear and such that $(xy)^{*}=(yx)^{*}$.
    2. An involutive Banach algebra is a Banach algebra equipped with an involution.

### isometry

    A linear isometry between normed vector spaces is a linear map preserving norm.

## K

### Krein–Milman

    The Krein–Milman theorem states: a nonempty compact convex subset of a locally convex space has an extremal point.

## L

### Locally convex algebra

    A locally convex algebra is an algebra whose underlying vector space is a locally convex space and whose multiplication is continuous with respect to the locally convex space topology.

## N

### nondegenerate

    A representation $(\pi ,V)$ of an algebra $A$ is said to be nondegenerate if for each vector $v\in V$, there is an element $a\in A$ such that $\pi (a)v\neq 0$.

### noncommutative

    1. noncommutative integration
    2. noncommutative torus

### norm

    1. A norm on a vector space X is a real-valued function $\|\cdot \|:X\to \mathbb {R}$ such that for each scalar $a$ and vectors $x,y$ in $X$, (1) $\|ax\|=|a|\|x\|$, (2) (triangular inequality) $\|x+y\|\leq \|x\|+\|y\|$ and (3) $\|x\|\geq 0$ where the equality holds only for $x=0$.
    2. A normed vector space is a real or complex vector space equipped with a norm $\|\cdot \|$. It is a metric space with the distance function $d(x,y)=\|x-y\|$.

### nuclear

    See nuclear operator.

## O

### one

    A one parameter group of a unital Banach algebra A is a continuous group homomorphism from $(\mathbb {R} ,+)$ to the unit group of A.

### orthonormal

    1. A subset S of a Hilbert space is orthonormal if, for each u, v in the set, $\langle u,v\rangle$ = 0 when $u\neq v$ and $=1$ when $u=v$.
    2. An orthonormal basis is a maximal orthonormal set (note: it is *not* necessarily a vector space basis.)

### orthogonal

    1. Given a Hilbert space H and a closed subspace M, the orthogonal complement of M is the closed subspace $M^{\bot }=\{x\in H|\langle x,y\rangle =0,y\in M\}$.
    2. In the notations above, the orthogonal projection $P$ onto M is a (unique) bounded operator on H such that $P^{2}=P,P^{*}=P,\operatorname {im} (P)=M,\operatorname {ker} (P)=M^{\bot }.$

## P

### Parseval

    Parseval's identity states: given an orthonormal basis S in a Hilbert space, $\|x\|^{2}=\sum _{u\in S}|\langle x,u\rangle |^{2}$.

### positive

    A linear functional $\omega$ on an involutive Banach algebra is said to be positive if $\omega (x^{*}x)\geq 0$ for each element $x$ in the algebra.

## Q

### quasitrace

    Quasitrace.

## R

### Radon

    See Radon measure.

### Riesz decomposition

    Riesz decomposition.

### Riesz's lemma

    Riesz's lemma.

### reflexive

    A reflexive space is a topological vector space such that the natural map from the vector space to the second (topological) dual is an isomorphism.

### resolvent

    The resolvent of an element x of a unital Banach algebra is the complement in $\mathbb {C}$ of the spectrum of x.

## S

### self-adjoint

    A self-adjoint operator is a bounded operator whose adjoint is itself.

### separable

    A separable Hilbert space is a Hilbert space admitting a finite or countable orthonormal basis.

### spectrum

    1. The spectrum of an element x of a unital Banach algebra is the set of complex numbers $\lambda$ such that $x-\lambda$ is not invertible.
    2. The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to $\mathbb {C}$) on the algebra.

### spectral

    1. The spectral radius of an element x of a unital Banach algebra is $sup_λ|λ|{\textstyle$ \sup _{\lambda }|\lambda |} where the sup is over the spectrum of x.
    2. The spectral mapping theorem states: if x is an element of a unital Banach algebra and f is a holomorphic function in a neighborhood of the spectrum $\sigma (x)$ of x, then $f(\sigma (x))=\sigma (f(x))$, where $f(x)$ is an element of the Banach algebra defined via the Cauchy's integral formula.

### state

    A state is a positive linear functional of norm one.

## T

### tensor product

    See topological tensor product. Note it is still somewhat of an open problem to define or work out a correct tensor product of topological vector spaces, including Banach spaces.

### topological

    A topological vector space is a vector space equipped with a topology such that (1) the topology is Hausdorff and (2) the addition $(x,y)\mapsto x+y$ as well as scalar multiplication $(\lambda ,x)\mapsto \lambda x$ are continuous.

## U

### unbounded operator

    An unbounded operator is a partially defined linear operator, usually defined on a dense subspace.

### uniform boundedness principle

    The uniform boundedness principle states: given a set of operators between Banach spaces, if $sup_T|Tx|<∞{\textstyle$ \sup _{T}|Tx|<\infty }, sup over the set, for each x in the Banach space, then $sup_T‖T‖<∞{\textstyle$ \sup _{T}\|T\|<\infty }.

### unitary

    1. A unitary operator between Hilbert spaces is an invertible bounded linear operator such that the inverse is the adjoint of the operator.
    2. Two representations $(\pi _{1},H_{1}),(\pi _{2},H_{2})$ of an involutive Banach algebra A on Hilbert spaces $H_{1},H_{2}$ are said to be unitarily equivalent if there is a unitary operator $U:H_{1}\to H_{2}$ such that $\pi _{2}(x)U=U\pi _{1}(x)$ for each x in A.

## W

### W*

    A W*-algebra is a C*-algebra that admits a faithful representation on a Hilbert space such that the image of the representation is a von Neumann algebra.
