Glossary of functional analysis

Last updated September 20, 2021

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

*

*: *-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,

\sum _{u\in S}|\langle x,u\rangle |^{2}\leq \|x\|^{2}

,^[1]

where the equality holds if and only if S is an orthonormal basis; i.e., maximal orthonormal set.

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, $\|x\|^{2}=\sum _{u\in S}|\langle x,u\rangle |^{2}$ .^[1]
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
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 ${\textstyle \sup _{\lambda }|\lambda |}$ 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 ${\textstyle \sup _{T}|Tx|<\infty }$ , sup over the set, for each x in the Banach space, then ${\textstyle \sup _{T}\|T\|<\infty }$ .
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.

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References

1 2 Here, the part of the assertion is $\sum _{u\in S}\cdots$ is well-defined; i.e., when S is infinite, for countable totally ordered subsets $S'\subset S$ , $\sum _{u\in S'}\cdots$ is independent of $S'$ and $\sum _{u\in S}\cdots$ denotes the common value.

Connes, Alain (1994), Non-commutative geometry , Boston, MA: Academic Press, ISBN 978-0-12-185860-5
Bourbaki, Espaces vectoriels topologiques
Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
M. Takesaki, Theory of Operator Algebras I, Springer, 2001, 2nd printing of the first edition 1979.
Yoshida, Kôsaku (1980), Functional Analysis (sixth ed.), Springer

Glossary of functional analysis

Contents

*

A

B

C

D

F

G

H

I

K

L

N

O

P

Q

R

S

T

U

W

Related Research Articles

References

Further reading