Glossary of functional analysis
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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 [math]\displaystyle{ T: H_1 \to H_2 }[/math] between Hilbert spaces is the bounded linear operator [math]\displaystyle{ T^* : H_2 \to H_1 }[/math] such that [math]\displaystyle{ \langle Tx, y \rangle = \langle x, T^* y \rangle }[/math] for each [math]\displaystyle{ x \in H_1, y \in H_2 }[/math].
- approximate identity
- In a not-necessarily-unital Banach algebra, an approximate identity is a sequence or a net [math]\displaystyle{ \{ u_i \} }[/math] of elements such that [math]\displaystyle{ u_i x \to x, x u_i \to x }[/math] as [math]\displaystyle{ i \to \infty }[/math] 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 [math]\displaystyle{ U_i }[/math] is a sequence of open dense subsets, then [math]\displaystyle{ \cap_1^{\infty} U_i }[/math] 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
- [math]\displaystyle{ \|x y \| \le \|x\| \|y\| }[/math] for every [math]\displaystyle{ x, y }[/math] in the algebra.
- [math]\displaystyle{ \sum_{u \in S} |\langle x, u \rangle|^2 \le \|x\|^2 }[/math],[1]
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 [math]\displaystyle{ x, y }[/math] in an inner-product space,
- [math]\displaystyle{ |\langle x, y \rangle| \le \|x\| \|y\| }[/math].
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 [math]\displaystyle{ \omega }[/math] on an involutive algebra is faithful if [math]\displaystyle{ \omega(x^*x) \ne 0 }[/math] for each nonzero element [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ A }[/math] with spectrum [math]\displaystyle{ \Omega(A) }[/math] is the algebra homomorphism [math]\displaystyle{ F: A \to C_0(\Omega(A)) }[/math], where [math]\displaystyle{ C_0(X) }[/math] denotes the algebra of continuous functions on [math]\displaystyle{ X }[/math] vanishing at infinity, that is given by [math]\displaystyle{ F(x)(\omega) = \omega(x) }[/math]. It is a *-preserving isometric isomorphism if [math]\displaystyle{ A }[/math] is a commutative C*-algebra.
- Grothendieck
- Grothendieck's inequality.
H
- Hahn–Banach
- The Hahn–Banach theorem states: given a linear functional [math]\displaystyle{ \ell }[/math] on a subspace of a complex vector space V, if the absolute value of [math]\displaystyle{ \ell }[/math] 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 [math]\displaystyle{ T }[/math] on a Hilbert space is [math]\displaystyle{ \sum_i \|T e_i \|^2 }[/math] where [math]\displaystyle{ \{ e_i \} }[/math] 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 [math]\displaystyle{ T : H_1 \to H_2 }[/math] is the integer [math]\displaystyle{ \operatorname{dim}(\operatorname{ker}(T^*)) - \operatorname{dim}(\operatorname{ker}(T)) }[/math].
- 2. The Atiyah–Singer index theorem.
- index group
- The index group of a unital Banach algebra is the quotient group [math]\displaystyle{ G(A)/G_0(A) }[/math] where [math]\displaystyle{ G(A) }[/math] is the unit group of A and [math]\displaystyle{ G_0(A) }[/math] the identity component of the group.
- inner product
- 1. An inner product on a real or complex vector space [math]\displaystyle{ V }[/math] is a function [math]\displaystyle{ \langle \cdot, \cdot \rangle : V \times V \to \mathbb{R} }[/math] such that for each [math]\displaystyle{ v, w \in V }[/math], (1) [math]\displaystyle{ x \mapsto \langle x, v \rangle }[/math] is linear and (2) [math]\displaystyle{ \langle v, w \rangle = \overline{\langle w, v\rangle} }[/math] 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 [math]\displaystyle{ A \to A, \, x \mapsto x^* }[/math] that is conjugate-linear and such that [math]\displaystyle{ (xy)^* = (yx)^* }[/math].
- 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 [math]\displaystyle{ (\pi, V) }[/math] of an algebra [math]\displaystyle{ A }[/math] is said to be nondegenerate if for each vector [math]\displaystyle{ v \in V }[/math], there is an element [math]\displaystyle{ a \in A }[/math] such that [math]\displaystyle{ \pi(a) v \ne 0 }[/math].
- noncommutative
- 1. noncommutative integration
- 2. noncommutative torus
- norm
- 1. A norm on a vector space X is a real-valued function [math]\displaystyle{ \| \cdot \| : X \to \mathbb{R} }[/math] such that for each scalar [math]\displaystyle{ a }[/math] and vectors [math]\displaystyle{ x, y }[/math] in [math]\displaystyle{ X }[/math], (1) [math]\displaystyle{ \| ax\| = |a| \| x \| }[/math], (2) (triangular inequality) [math]\displaystyle{ \| x + y \| \le \| x \| + \| y \| }[/math] and (3) [math]\displaystyle{ \| x \| \ge 0 }[/math] where the equality holds only for [math]\displaystyle{ x = 0 }[/math].
- 2. A normed vector space is a real or complex vector space equipped with a norm [math]\displaystyle{ \| \cdot \| }[/math]. It is a metric space with the distance function [math]\displaystyle{ d(x, y) = \| x - y \| }[/math].
- nuclear
- See nuclear operator.
O
- one
- A one parameter group of a unital Banach algebra A is a continuous group homomorphism from [math]\displaystyle{ (\mathbb{R}, +) }[/math] 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, [math]\displaystyle{ \langle u, v \rangle }[/math] = 0 when [math]\displaystyle{ u \ne v }[/math] and [math]\displaystyle{ = 1 }[/math] when [math]\displaystyle{ u = v }[/math].
- 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 [math]\displaystyle{ M^{\bot} = \{ x \in H | \langle x, y \rangle = 0, y \in M \} }[/math].
- 2. In the notations above, the orthogonal projection [math]\displaystyle{ P }[/math] onto M is a (unique) bounded operator on H such that [math]\displaystyle{ P^2 = P, P^* = P, \operatorname{im}(P) = M, \operatorname{ker}(P) = M^{\bot}. }[/math]
P
- Parseval
- Parseval's identity states: given an orthonormal basis S in a Hilbert space, [math]\displaystyle{ \| x \|^2 = \sum_{u \in S} |\langle x, u \rangle|^2 }[/math].[1]
- positive
- A linear functional [math]\displaystyle{ \omega }[/math] on an involutive Banach algebra is said to be positive if [math]\displaystyle{ \omega(x^* x) \ge 0 }[/math] for each element [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ \mathbb{C} }[/math] 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 [math]\displaystyle{ \lambda }[/math] such that [math]\displaystyle{ x - \lambda }[/math] is not invertible.
- 2. The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to [math]\displaystyle{ \mathbb{C} }[/math]) on the algebra.
- spectral
- 1. The spectral radius of an element x of a unital Banach algebra is [math]\displaystyle{ \sup_{\lambda} |\lambda| }[/math] 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 [math]\displaystyle{ \sigma(x) }[/math] of x, then [math]\displaystyle{ f(\sigma(x)) = \sigma(f(x)) }[/math], where [math]\displaystyle{ f(x) }[/math] 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 [math]\displaystyle{ (x, y) \mapsto x + y }[/math] as well as scalar multiplication [math]\displaystyle{ (\lambda, x) \mapsto \lambda x }[/math] 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 [math]\displaystyle{ \sup_T |Tx| \lt \infty }[/math], sup over the set, for each x in the Banach space, then [math]\displaystyle{ \sup_T \|T\| \lt \infty }[/math].
- 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 [math]\displaystyle{ (\pi_1, H_1), (\pi_2, H_2) }[/math] of an involutive Banach algebra A on Hilbert spaces [math]\displaystyle{ H_1, H_2 }[/math] are said to be unitarily equivalent if there is a unitary operator [math]\displaystyle{ U: H_1 \to H_2 }[/math] such that [math]\displaystyle{ \pi_2(x) U = U \pi_1(x) }[/math] 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
- ↑ 1.0 1.1 Here, the part of the assertion is [math]\displaystyle{ \sum_{u \in S} \cdots }[/math] is well-defined; i.e., when S is infinite, for countable totally ordered subsets [math]\displaystyle{ S' \subset S }[/math], [math]\displaystyle{ \sum_{u \in S'} \cdots }[/math] is independent of [math]\displaystyle{ S' }[/math] and [math]\displaystyle{ \sum_{u \in S} \cdots }[/math] denotes the common value.
- Connes, Alain (1994), Non-commutative geometry, Boston, MA: Academic Press, ISBN 978-0-12-185860-5, https://archive.org/details/noncommutativege0000conn
- Bourbaki, Espaces vectoriels topologiques
- 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
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