Glossary of functional analysis

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See also: List of functional analysis topics

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, glossary of real and complex 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.

Anderson–Kadec

The Anderson–Kadec theorem says a separable infinite-dimensional Fréchet space is isomorphic to ${\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}$.

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 $⟨Tx,y⟩=⟨x,T^{*}y⟩$ 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 $i\to \mathrm{\infty }$ 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 $\underset{1}{\overset{\mathrm{\infty }}{\cap }}{U}_i$ 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

$\Vert xy\Vert \le \Vert x\Vert \Vert y\Vert$ for every $x,y$ in the algebra.

3.  A Banach disc is a continuous linear image of a unit ball in a Banach space.

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

1.  A barrel in a topological vector space is a subset that is closed, convex, balanced and absorbing.

2.  A topological vector space is barrelled if every barrel 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,

$\sum _{u\in S}|⟨x,u⟩{|}^2\le \Vert x{\Vert }^2$,

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

bipolar

bipolar theorem.

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

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

Borel

Borel functional calculus

C

c

c space.

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,

$|⟨x,y⟩|\le \Vert x\Vert \Vert y\Vert$.

closed

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

2.  A closed operator is a linear operator whose graph is closed.

3.  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

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 $S^{\prime }$.

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.

Connes

Connes fusion.

C*

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

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

dilation

dilation (operator theory).

direct

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

Douglas

Douglas' lemma

Dunford

Dunford–Schwartz theorem

dual

1.  The continuous dual of a topological vector space is the vector space of all the continuous linear functionals on the space.

2.  The algebraic dual of a topological vector space is the dual vector space of the underlying vector space.

E

Eidelheit

A theorem of Eidelheit.

essentially selfadjoint

An essentially selfadjoint operator.

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

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 $\mathrm{\Omega }(A)$ is the algebra homomorphism $F:A\to C_0(\mathrm{\Omega }(A))$, 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

1.  Grothendieck's inequality.

2.  Grothendieck's factorization theorem.

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.

Heine

A topological vector space is said to have the Heine–Borel property if every closed and bounded subset is compact. Riesz's lemma says a Banach space with the Heine–Borel property must be finite-dimensional.

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 $\sum _i\Vert T{e}_i{\Vert }^2$ 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 $\dim (\ker (T^{*}))-\dim (\ker (T))$.

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.

infra-barrelled

infra-barrelled

inner product

1.  An inner product on a real or complex vector space $V$ is a function $⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{ℝ}$ such that for each $v,w\in V$, (1) $x\mapsto ⟨x,v⟩$ is linear and (2) $⟨v,w⟩=\overline{⟨w,v⟩}$ 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

Kato–Rellich

The Kato–Rellich theorem

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.

Krein–Smulian

Krein–Smulian theorem

L

Linear

Linear Operators is a three-value book by Dunford and Schwartz.

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.

M

Mazur

Mazur–Ulam theorem.

Montel

Montel space.

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

1.  noncommutative integration

2.  noncommutative torus

norm

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

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

normal

An operator is normal if it and its adjoint commute.

nuclear

1.  nuclear operator.

2.  nuclear space.

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

1.  A subset S of a Hilbert space is orthonormal if, for each u, v in the set, $⟨u,v⟩$ = 0 when $u\ne v$ 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 $M^{\mathrm{\perp }}=\{x\in H|⟨x,y⟩=0,y\in M\}$.

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

P

Parseval

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

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.

predual

predual.

projection

An operator T is called a projection if it is an idempotent; i.e., $T^2=T$.

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.

Ryll-Nardzewski

Ryll-Nardzewski fixed-point theorem.

S

Schauder

Schauder basis.

Schatten

Schatten class

selection

Michael selection theorem.

self-adjoint

A self-adjoint operator is a bounded operator whose adjoint is itself. More generally, a closed densely defined operator is called self-adjoint if it coincides with the adjoint including the domain.

semi-reflexive

A locally convex space is called semi-reflexive space if the canonical map to the second continuous dual is surjective.

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 $\underset{\lambda }{\sup }|\lambda |$ 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.

Stone

Stone lemma.

symmetric

A linear operator T on a pre-Hilbert space is symmetric if $(Tx,y)=(x,Ty).$

T

tensor product

1.  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.

2.  A projective tensor product.

topological

1.  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.

2.  A linear map $f:E\to F$ is called a topological homomorphism if $f:E\to \mathrm{im}(f)$ is an open mapping.

3.  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

ultraweak

ultraweak topology.

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 $\underset{T}{\sup }|Tx|<\mathrm{\infty }$, sup over the set, for each x in the Banach space, then $\underset{T}{\sup }\Vert T\Vert <\mathrm{\infty }$.

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.

V

von Neumann

1.  A von Neumann algebra.

2.  von Neumann's theorem.

3.  Von Neumann's inequality.

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

• Bourbaki, Espaces vectoriels topologiques
• Connes, Alain (1994), Non-commutative geometry, Boston, MA: Academic Press, ISBN 978-0-12-185860-5, https://archive.org/details/noncommutativege0000conn 
• Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. 
• Template:Dunford Schwartz Linear Operators Part 1 General Theory
• Rudin, Walter (January 1, 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. https://archive.org/details/functionalanalys00rudi. 
• 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
• https://mathoverflow.net/questions/408415/takesaki-theorem-2-6