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.
*
[edit]- *
- *-homomorphism between involutive Banach algebras is an algebra homomorphism preserving *.
A
[edit]- 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 .
- 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
[edit]- 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.
- ,[1]
C
[edit]- 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 in an inner-product space,
- .
D
[edit]- 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
[edit]- Eidelheit
- A theorem of Eidelheit.
F
[edit]- 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
[edit]- 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
- 1. Grothendieck's inequality.
- 2. Grothendieck's factorization theorem.
H
[edit]- 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.
- 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 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
[edit]- 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
[edit]- 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
[edit]- 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
[edit]- Mazur
- Mazur–Ulam theorem.
- Montel
- Montel space.
N
[edit]- 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 .
- normal
- An operator is normal if it and its adjoint commute.
- nuclear
- See nuclear operator.
O
[edit]- one
- A one parameter group of a unital Banach algebra A is a continuous group homomorphism from 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, = 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
[edit]- 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.
- predual
- predual.
- projection
- An operator T is called a projection if it is an idempotent; i.e., .
Q
[edit]- quasitrace
- Quasitrace.
R
[edit]- 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 of the spectrum of x.
- Ryll-Nardzewski
- Ryll-Nardzewski fixed-point theorem.
S
[edit]- Schauder
- Schauder basis.
- Schatten
- Schatten class
- selection
- Michael selection theorem.
- 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.
- symmetric
- A linear operator T on a pre-Hilbert space is symmetric if
T
[edit]- 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 as well as scalar multiplication are continuous.
- 2. A linear map is called a topological homomorphism if is an open mapping.
- 3. A sequence is called topologically exact if it is an exact sequence on the underlying vector spaces and, moreover, each is a topological homomorphism.
U
[edit]- 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 , 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.
V
[edit]- von Neumann
- 1. A von Neumann algebra.
- 2. von Neumann's theorem.
- 3. Von Neumann's inequality.
W
[edit]- 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
[edit]- Bourbaki, Espaces vectoriels topologiques
- Connes, Alain (1994), Non-commutative geometry, Boston, MA: Academic Press, ISBN 978-0-12-185860-5
- Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
- Dunford, Nelson; Schwartz, Jacob T. (1988). Linear Operators. Pure and applied mathematics. Vol. 1. New York: Wiley-Interscience. ISBN 978-0-471-60848-6. OCLC 18412261.
- 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
[edit]- 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