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# List of functional analysis topics

This is a list of functional analysis topics, by Wikipedia page. [1]

## *-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings and, where is commutative and has the structure of an associative algebra over.

## Abelian von Neumann algebra

In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.

## Alain Connes

Alain Connes (born 1 April 1947) is a French mathematician, currently Professor at the Collège de France, IHÉS, Ohio State University and Vanderbilt University.

## Amenable group

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements.

## Approximation property

In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators.

## Arzelà–Ascoli theorem

The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence.

## Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

## Ba space

In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma.

## Baire category theorem

The Baire category theorem (BCT) is an important tool in general topology and functional analysis.

## Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Banach–Alaoglu theorem

In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.

## Banach–Mazur theorem

In mathematics, the Banach–Mazur theorem is a theorem of functional analysis.

## Barrelled space

In functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector.

## Basis function

In mathematics, a basis function is an element of a particular basis for a function space.

## Bernstein's theorem on monotone functions

In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line.

## Bicommutant

In algebra, the bicommutant of a subset S of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset.

## Borel functional calculus

In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope.

## Bornological space

In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and functions, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity.

## Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

## Bra–ket notation

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states.

## C*-algebra

C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.

## Calkin algebra

In functional analysis, the Calkin algebra, named after, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators.

## Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S are elements that satisfy a weaker condition.

## Closed graph theorem

In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs.

## Coherent states

In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator.

## Colombeau algebra

In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions.

## Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y. Such an operator is necessarily a bounded operator, and so continuous.

## Complex Mexican hat wavelet

In applied mathematics, the complex Mexican hat wavelet is a low-oscillation, complex-valued, wavelet for the continuous wavelet transform.

## Compression (functional analysis)

In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator where P_K: H \rightarrow K is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space.

## Continuous functional calculus

In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.

## Continuous linear extension

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space X by first defining a linear transformation \mathsf on a dense subset of X and then extending \mathsf to the whole space via the theorem below.

## Continuous wavelet

In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform.

## Continuous wavelet transform

In mathematics, a continuous wavelet transform (CWT) is used to divide a continuous-time function into wavelets.

## Daubechies wavelet

The Daubechies wavelets, based on the work of Ingrid Daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support.

In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.

## Density matrix

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.

## Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

## Direct integral

In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum.

## Discrete wavelet transform

In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled.

## Disk algebra

In functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions where D is the open unit disk in the complex plane C, f extends to a continuous function on the closure of D. That is, where H^\infty(\mathbf) denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space).

## Dual pair

In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear map to the base field.

## Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

## Dual topology

In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.

## Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

## Essential spectrum

In mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## F-space

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d: V × V → R so that.

## Fixed-point theorems in infinite-dimensional spaces

In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem.

## Fock space

The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space.

## Fock state

In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta).

## Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.

## Fredholm operator

In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations.

## Free probability

Free probability is a mathematical theory that studies non-commutative random variables.

## Friedrichs extension

In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator.

## Fuglede's theorem

In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

## Functional calculus

In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators.

## Gelfand representation

In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings.

## Gelfand–Naimark theorem

In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space.

## Gelfand–Naimark–Segal construction

In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A (called states).

## Gram–Schmidt process

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product.

## Haar wavelet

In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis.

## Hahn–Banach theorem

In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.

## Hardy space

In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane.

## Heisenberg picture

In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.

## Hellinger–Toeplitz theorem

In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product \langle \cdot | \cdot \rangle is bounded.

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding adjoint operator.

## Hermitian hat wavelet

The Hermitian hat wavelet is a low-oscillation, complex-valued wavelet.

## Hermitian wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform.

## Hilbert matrix

In linear algebra, a Hilbert matrix, introduced by, is a square matrix with entries being the unit fractions For example, this is the 5 &times; 5 Hilbert matrix: 1 & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \end.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Hilbert–Pólya conjecture

In mathematics, the Hilbert–Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory.

## Hugo Steinhaus

Władysław Hugo Dionizy Steinhaus (January 14, 1887 – February 25, 1972) was a Jewish-Polish mathematician and educator.

## Hurwitz's theorem (composition algebras)

In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form.

## Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

## Invariant subspace

In mathematics, an invariant subspace of a linear mapping T: V &rarr; V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) &sube; W.

## John von Neumann

John von Neumann (Neumann János Lajos,; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath.

## Krein–Milman theorem

In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about convex sets in topological vector spaces.

## Legendre polynomials

In mathematics, Legendre functions are solutions to Legendre's differential equation: They are named after Adrien-Marie Legendre.

## List of mathematical topics in quantum theory

This is a list of mathematical topics in quantum theory, by Wikipedia page.

## Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces.

## Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

## Mackey topology

In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual.

## Mathematical formulation of quantum mechanics

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

## Matrix norm

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

## Measure of non-compactness

In functional analysis, two measures of non-compactness are commonly used; these associate numbers to sets in such a way that compact sets all get the measure 0, and other sets get measures that are bigger according to "how far" they are removed from compactness.

## Mercer's theorem

In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions.

## Mexican hat wavelet

In mathematics and numerical analysis, the Ricker wavelet is the negative normalized second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite function.

## Min-max theorem

In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant&ndash;Fischer&ndash;Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces.

## Montel space

In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space in which an analog of Montel's theorem holds.

## Morlet wavelet

In mathematics, the Morlet wavelet (or Gabor wavelet) "The Gabor kernel satisfies the admissibility condition for wavelets, thus being suited for multi-resolution analysis.

## Nest algebra

In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context.

## Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense).

## Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

## Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

## Normal matrix

In mathematics, a complex square matrix is normal if where is the conjugate transpose of.

## Normal operator

In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN*.

## Normed vector space

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.

## Observable

In physics, an observable is a dynamic variable that can be measured.

## Open mapping theorem (functional analysis)

In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.

## Operator (physics)

In physics, an operator is a function over a space of physical states to another space of physical states.

## Operator algebra

In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings.

## Operator norm

In mathematics, the operator norm is a means to measure the "size" of certain linear operators.

## Operator theory

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators.

## Operator topologies

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X.

## Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement.

## Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.

## Orthogonalization

In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.

## Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

## Parallelogram law

In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry.

## Parseval's identity

In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function.

## Polar set

In functional analysis and related areas of mathematics the polar set of a given subset of a vector space is a certain set in the dual space.

## Polar topology

In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a dual pair.

## Polynomially reflexive space

In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space.

## Positive element

In mathematics, especially functional analysis, a self-adjoint (or Hermitian) element A of a C*-algebra \mathcal is called positive if its spectrum \sigma(A) consists of non-negative real numbers.

## Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, ≤) is a linear functional f on V so that for all positive elements v of V, that is v≥0, it holds that In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements.

## Predual

In mathematics, the predual of an object D is an object P whose dual space is D. For example, the predual of the space of bounded operators is the space of trace class operators.

## Probability

Probability is the measure of the likelihood that an event will occur.

## Quantum logic

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account.

## Quantum operation

In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo.

## Quantum state

In quantum physics, quantum state refers to the state of an isolated quantum system.

## Rayleigh quotient

In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient R(M, x), is defined as: For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x^ to the usual transpose x'.

## Reflexive operator algebra

In functional analysis, a reflexive operator algebra A is an operator algebra that has enough invariant subspaces to characterize it.

## Reflexive space

In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space.

## Reproducing kernel Hilbert space

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional.

## Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem.

## Rigged Hilbert space

In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis.

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.

## Semi-Hilbert space

In mathematics, a semi-Hilbert space is a generalization of a Hilbert space in functional analysis, in which, roughly speaking, the inner product is required only to be positive semi-definite rather than positive definite, so that it gives rise to a seminorm rather than a vector space norm.

## Shift operator

In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation.

## Singular value

In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator acting between Hilbert spaces X and Y, are the square roots of the eigenvalues of the non-negative self-adjoint operator (where T* denotes the adjoint of T).

## Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.

## Spectral density

The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal.

In mathematics, the spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its eigenvalues (i.e. supremum among the absolute values of the elements in its spectrum).

## Spectral theorem

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).

## Spectral theory

In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.

## Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix.

## Spectrum of a C*-algebra

In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces.

## Stefan Banach

Stefan Banach (30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.

## Stone's theorem on one-parameter unitary groups

In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families of unitary operators that are strongly continuous, i.e., and are homomorphisms, i.e., Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

## Stone–von Neumann theorem

In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.

## Stone–Weierstrass theorem

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

## Strong operator topology

In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form T\mapsto\|Tx\|, as x varies in H. Equivalently, it is the coarsest topology such that the evaluation maps T\mapsto Tx (taking values in H) are continuous for each fixed x in H. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U(T_0,x,\epsilon).

## Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

## Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps where R &times; R carries the product topology.

## Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

## Trace class

In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis.

## Tsirelson space

In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ''p'' space nor a ''c''0 space can be embedded.

## Ultrastrong topology

In functional analysis, the ultrastrong topology, or &sigma;-strong topology, or strongest topology on the set B(H) of bounded operators on a Hilbert space is the topology defined by the family of seminorms p_\omega(x).

## Ultraweak topology

In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B(H) of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B*(H) of B(H), the trace class operators on H. In other words it is the weakest topology such that all elements of the predual are continuous (when considered as functions on B(H)).

## Uniform boundedness principle

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.

## Uniform norm

In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.

## Unit sphere

In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.

## Unitary matrix

In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.

## Universal C*-algebra

In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations.

## Von Neumann algebra

In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator.

## Von Neumann bicommutant theorem

In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set.

## Von Neumann conjecture

In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators.

## Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero.

## Weak operator topology

In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is continuous for any vectors x and y in the Hilbert space.

## Weak topology

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space.

## Wightman axioms

In physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Lars Gårding and Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory.

## References

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