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.

See also: List of Banach spaces.

*

 * -homomorphism between involutive Banach algebras is an algebra homomorphism preserving *.
 * -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 $u_i x \to x, x u_i \to x$ 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: A Banach space is a normed vector space that is complete as a metric space. x y \

balanced: A subset S of a vector space over real or complex numbers is balanced if $\lambda S \subset S$ for every scalar $\lambda$ of length at most one.

barrel: A barrel in a topological vector space is a subset that is closed, convex, balanced and absorbing. A topological vector space is barrelled if every barrell is a neighborhood of zero (that is, contains an open neighborhood of zero).

Bessel: Bessel's inequality states: given an orthonormal set S and a vector x in a Hilbert space,
 * $\sum_{u \in S}

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

bornological: A bornological space.

Birkhoff orthogonality: x + \lambda y \

Borel: Borel functional calculus

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: \langle x, y \rangle

closed: The closed graph theorem states that a linear operator between Banach spaces is continuous (bounded) if and only if it has closed graph. A closed operator is a linear operator whose graph is closed. The closed range theorem says that a densely defined closed operator has closed image (range) if and only if the transpose of it has closed image.

commutant: 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 $S'$. 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*: 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.

dual: The continuous dual of a topological vector space is the vector space of all the continuous linear functionals on the space. The algebraic dual of a topological vector space is the dual vector space of the underlying vector space.

F
factor: A factor is a von Neumann algebra with trivial center.

faithful: A linear functional $\omega$ on an involutive algebra is faithful if $\omega(x^*x) \ne 0$ 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: The Gelfand–Mazur theorem states that a Banach algebra that is a division ring is the field of complex numbers. 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: A Hilbert space is an inner product space that is complete as a metric space. In the Tomita–Takesaki theory, a (left or right) Hilbert algebra is a certain algebra with an involution.

Hilbert–Schmidt: T e_i \ A Hilbert–Schmidt operator is a bounded operator with finite Hilbert–Schmidt norm.

I
index: The index of a Fredholm operator $T : H_1 \to H_2$ is the integer $\operatorname{dim}(\operatorname{ker}(T^*)) - \operatorname{dim}(\operatorname{ker}(T))$. 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: 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. An inner product space is a vector space equipped with an inner product.

involution: An involution of a Banach algebra A is an isometric endomorphism $A \to A, \, x \mapsto x^*$ that is conjugate-linear and such that $(xy)^* = (yx)^*$. 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
Köthe: A Köthe sequence space. For now, see https://mathoverflow.net/questions/361048/on-k%C3%B6the-sequence-spaces

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 $\pi(a) v \ne 0$.

noncommutative: noncommutative integration noncommutative torus

norm: \cdot \ \cdot \

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.

open: The open mapping theorem says a surjective continuous linear operator between Banach spaces is an open mapping.

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

orthogonal: \langle x, y \rangle = 0, y \in M \}$. 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: x \

positive: A linear functional $\omega$ on an involutive Banach algebra is said to be positive if $\omega(x^* x) \ge 0$ 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
Schauder: Schauder basis.

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: 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. The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to $\mathbb{C}$) on the algebra.

spectral: \lambda 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. A projective tensor product.

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. A linear map $f: E \to F$ is called a topological homomorphism if $f : E \to \operatorname{im}(f)$ is an open mapping. A sequence $\cdots \to E_{n -1} \to E_n \to E_{n+1} \to \cdots$ is called topologically exact if it is an exact sequence on the underlying vector spaces and, moreover, each $ E_n \to E_{n+1}$ is a topological homomorphism.

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

uniform boundedness principle: Tx

unitary: A unitary operator between Hilbert spaces is an invertible bounded linear operator such that the inverse is the adjoint of the operator. 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.

V
von Neumann: A von Neumann algebra. A von Neumann's theorem.

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.