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.

A

*
*-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 between Hilbert spaces is the bounded linear operator such that for each .
approximate identity
In a not-necessarily-unital Banach algebra, an approximate identity is a sequence or a net of elements such that as 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 is a sequence of open dense subsets, then 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
for every in the algebra.
Bessel
Bessel's inequality states: given an orthonormal set S and a vector x in a Hilbert space,
,[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 that maps a closed ball to a closed ball.
Birkhoff orthogonality
Two vectors x and y in a normed linear space are said to be Birkhoff orthogonal if 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 in an inner-product space,
.
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 .
2.  The von Neumann double commutant theorem states that a nondegenerate *-algebra of operators on a Hilbert space is a von Neumann algebra; i.e., .
compact
A compact operator is a linear operator between Banach spaces that maps a closed ball to a compact set.
C*
A C* algebra is an involutive Banach algebra satisfying .
convex
A locally convex space is a topological vector space whose topology is generated by convex subsets.
cyclic
Given a representation of a Banach algebra , a cyclic vector is a vector such that is dense in .

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 on an involutive algebra is faithful if for each nonzero element 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 with spectrum is the algebra homomorphism , where denotes the algebra of continuous functions on vanishing at infinity, that is given by . It is a *-preserving isometric isomorphism if is a commutative C*-algebra.
Grothendieck
Grothendieck's inequality.

H

Hahn–Banach
The Hahn–Banach theorem states: given a linear functional on a subspace of a complex vector space V, if the absolute value of 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 on a Hilbert space is where 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 is the integer .
2.  The Atiyah–Singer index theorem.
index group
The index group of a unital Banach algebra is the quotient group where is the unit group of A and the identity component of the group.
inner product
1.  An inner product on a real or complex vector space is a function such that for each , (1) is linear and (2) 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 that is conjugate-linear and such that .
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 of an algebra is said to be nondegenerate if for each vector , there is an element such that .
noncommutative
1.  noncommutative integration
2.  noncommutative torus
norm
1.  A norm on a vector space X is a real-valued function such that for each scalar and vectors in , (1) , (2) (triangular inequality) and (3) where the equality holds only for .
2.  A normed vector space is a real or complex vector space equipped with a norm . It is a metric space with the distance function .
nuclear
See nuclear operator.

O

one
A one parameter group of a unital Banach algebra A is a continuous group homomorphism from 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, = 0 when and when .
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 .
2.  In the notations above, the orthogonal projection onto M is a (unique) bounded operator on H such that

P

Parseval
Parseval's identity states: given an orthonormal basis S in a Hilbert space, .[1]
positive
A linear functional on an involutive Banach algebra is said to be positive if for each element 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 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 such that is not invertible.
2.  The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to ) on the algebra.
spectral
1.  The spectral radius of an element x of a unital Banach algebra is 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 of x, then , where 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 as well as scalar multiplication are continuous.

U

unbounded operator
An unbounded operator is a partially defined linear operator, usually a bounded operator on some dense subspace.
uniform boundedness principle
The uniform boundedness principle states: given a set of operators between Banach spaces, if , sup over the set, for each x in the Banach space, then .
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 of an involutive Banach algebra A on Hilbert spaces are said to be unitarily equivalent if there is a unitary operator such that 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

  1. Here, the part of the assertion is is well-defined; i.e., when S is infinite, for countable totally ordered subsets , is independent of and 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

Further reading


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