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.

*

*: *-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: 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 ${\textstyle \sup _{\lambda }|\lambda |}$ 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.

References

^ ^a ^b 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. Vol. 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

*

A

B

C

D

F

G

H

I

K

L

N

O

P

Q

R

S

T

U

W

References

Further reading