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 *.

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

approximate identity

In a not-necessarily-unital Banach algebra, an approximate identity is a sequence or a net ${\displaystyle \{u_{i}\}}$ of elements such that ${\displaystyle u_{i}x\to x,xu_{i}\to x}$ as ${\displaystyle i\to \infty }$ 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 ${\displaystyle U_{i}}$ is a sequence of open dense subsets, then ${\displaystyle \cap _{1}^{\infty }U_{i}}$ 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

${\displaystyle \|xy\|\leq \|x\|\|y\|}$ for every ${\displaystyle x,y}$ in the algebra.

Bessel

Bessel's inequality states: given an orthonormal set S and a vector x in a Hilbert space,

${\displaystyle \sum _{u\in S}|\langle x,u\rangle |^{2}\leq \|x\|^{2}}$,^[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 ${\displaystyle \|x+\lambda y\|\geq \|x\|}$ 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 ${\displaystyle x,y}$ in an inner-product space,

${\displaystyle |\langle x,y\rangle |\leq \|x\|\|y\|}$.

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 ${\displaystyle S'}$.

2.  The von Neumann double commutant theorem states that a nondegenerate *-algebra ${\displaystyle {\mathfrak {M}}}$ of operators on a Hilbert space is a von Neumann algebra if and only if ${\displaystyle {\mathfrak {M}}''={\mathfrak {M}}}$.

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 ${\displaystyle \|x^{*}x\|=\|x^{*}\|\|x\|}$.

convex

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

cyclic

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

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 ${\displaystyle \omega }$ on an involutive algebra is faithful if ${\displaystyle \omega (x^{*}x)\neq 0}$ for each nonzero element ${\displaystyle x}$ 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 ${\displaystyle A}$ with spectrum ${\displaystyle \Omega (A)}$ is the algebra homomorphism ${\displaystyle F:A\to C_{0}(\Omega (A))}$, where ${\displaystyle C_{0}(X)}$ denotes the algebra of continuous functions on ${\displaystyle X}$ vanishing at infinity, that is given by ${\displaystyle F(x)(\omega )=\omega (x)}$. It is a *-preserving isometric isomorphism if ${\displaystyle A}$ is a commutative C*-algebra.

Grothendieck

Grothendieck's inequality.

H

Hahn–Banach

The Hahn–Banach theorem states: given a linear functional ${\displaystyle \ell }$ on a subspace of a complex vector space V, if the absolute value of ${\displaystyle \ell }$ 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 ${\displaystyle T}$ on a Hilbert space is ${\displaystyle \sum _{i}\|Te_{i}\|^{2}}$ where ${\displaystyle \{e_{i}\}}$ 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 ${\displaystyle T:H_{1}\to H_{2}}$ is the integer ${\displaystyle \operatorname {dim} (\operatorname {ker} (T^{*}))-\operatorname {dim} (\operatorname {ker} (T))}$.

2.  The Atiyah–Singer index theorem.

index group

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

inner product

1.  An inner product on a real or complex vector space ${\displaystyle V}$ is a function ${\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to \mathbb {R} }$ such that for each ${\displaystyle v,w\in V}$, (1) ${\displaystyle x\mapsto \langle x,v\rangle }$ is linear and (2) ${\displaystyle \langle v,w\rangle ={\overline {\langle w,v\rangle }}}$ 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 ${\displaystyle A\to A,\,x\mapsto x^{*}}$ that is conjugate-linear and such that ${\displaystyle (xy)^{*}=(yx)^{*}}$.

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

noncommutative

1.  noncommutative integration

2.  noncommutative torus

norm

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

2.  A normed vector space is a real or complex vector space equipped with a norm ${\displaystyle \|\cdot \|}$. It is a metric space with the distance function ${\displaystyle d(x,y)=\|x-y\|}$.

nuclear

See nuclear operator.

O

one

A one parameter group of a unital Banach algebra A is a continuous group homomorphism from ${\displaystyle (\mathbb {R} ,+)}$ 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, ${\displaystyle \langle u,v\rangle }$ = 0 when ${\displaystyle u\neq v}$ and ${\displaystyle =1}$ when ${\displaystyle u=v}$.

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 ${\displaystyle M^{\bot }=\{x\in H|\langle x,y\rangle =0,y\in M\}}$.

2.  In the notations above, the orthogonal projection ${\displaystyle P}$ onto M is a (unique) bounded operator on H such that ${\displaystyle P^{2}=P,P^{*}=P,\operatorname {im} (P)=M,\operatorname {ker} (P)=M^{\bot }.}$

P

Parseval

Parseval's identity states: given an orthonormal basis S in a Hilbert space, ${\displaystyle \|x\|^{2}=\sum _{u\in S}|\langle x,u\rangle |^{2}}$.^[1]

positive

A linear functional ${\displaystyle \omega }$ on an involutive Banach algebra is said to be positive if ${\displaystyle \omega (x^{*}x)\geq 0}$ for each element ${\displaystyle x}$ 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 ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \lambda }$ such that ${\displaystyle x-\lambda }$ is not invertible.

2.  The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to ${\displaystyle \mathbb {C} }$) 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 ${\displaystyle \sigma (x)}$ of x, then ${\displaystyle f(\sigma (x))=\sigma (f(x))}$, where ${\displaystyle f(x)}$ 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 ${\displaystyle (x,y)\mapsto x+y}$ as well as scalar multiplication ${\displaystyle (\lambda ,x)\mapsto \lambda x}$ 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 ${\displaystyle (\pi _{1},H_{1}),(\pi _{2},H_{2})}$ of an involutive Banach algebra A on Hilbert spaces ${\displaystyle H_{1},H_{2}}$ are said to be unitarily equivalent if there is a unitary operator ${\displaystyle U:H_{1}\to H_{2}}$ such that ${\displaystyle \pi _{2}(x)U=U\pi _{1}(x)}$ 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

• 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

Further reading

• Antony Wassermann's lecture notes at http://iml.univ-mrs.fr/~wasserm/
• Jacob Lurie's lecture notes on a von Neumann algebra at https://www.math.ias.edu/~lurie/261y.html