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.
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 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.
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 if and only if .
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 .
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.
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.
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 .
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 .
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 defined on a 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
^
abHere, 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.