Hilbert space

• Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
  
• Inner product space
  Inner product space
  In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
  
  
• Hilbert space
  Hilbert space
  The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
  
  
• Parallelogram law
  Parallelogram law
  In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals...
  
  
• Orthonormal basis
  Orthonormal basis
  In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
  
  
• Orthogonalization
• Orthogonal complement
• Gram–Schmidt process
  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...
  
  
• Legendre polynomials
• Matrices
  • Orthogonal matrix
    Orthogonal matrix
    In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
    
    
  • Unitary matrix
  • Normal matrix
    Normal matrix
    A complex square matrix A is a normal matrix ifA^*A=AA^* \ where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.If A is a real matrix, then A*=AT...
    
    , normal operator
    Normal operator
    In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operatorN:H\to Hthat commutes with its hermitian adjoint N*: N\,N^*=N^*N....
    
    
  • Symmetric matrix
  • Hermitian operator self-adjoint operator
    Self-adjoint operator
    In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
    
    , Hermitian adjoint
    Hermitian adjoint
    In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...
    
    
  • Eigenvector, eigenvalue, eigenfunction
    Eigenfunction
    In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
    
    
  • Diagonal matrix
    Diagonal matrix
    In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
    
    
  • Shift operator
    Shift operator
    In mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator....
    
    
  • Hilbert matrix
• Normal vector
• Parseval's identity
  Parseval's identity
  In mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is thePythagorean theorem for inner-product spaces....
  
  
• Riesz representation theorem
  Riesz representation theorem
  There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...
  
  
• Bra-ket notation
  Bra-ket notation
  Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
  
  
• Spectral theorem
  Spectral theorem
  In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
  
  , Spectral theory
  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. It is a result of studies of linear algebra and the solutions of...
  
  
• Positive-definite
• Rayleigh quotient
• Mercer's theorem
  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. This theorem, presented in , is one of the most notable results of the work of James Mercer...
  
  
• Reproducing kernel Hilbert space
  Reproducing kernel Hilbert space
  In functional analysis , a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels...
  
  
• Trace class
  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....
  
  
• Min-max theorem
  Min-max theorem
  In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces...
  
  
• Rigged Hilbert space
  Rigged Hilbert space
  In mathematics, a rigged Hilbert space is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense...
  
  
• Hellinger–Toeplitz theorem
  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 is bounded. By definition, an operator A is symmetric if \langle A x | y \rangle = \langle x | A y\rangle for all x, y in the domain of A...
  
  
• Direct integral
  Direct integral
  In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers...
  
  
• Semi-Hilbert space
  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.The...
  
  

Functional analysis, classic results

• Normed vector space
  Normed vector space
  In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
  
  
  • Unit ball
• Banach space
  Banach space
  In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
  
  
• Hahn–Banach theorem
  Hahn–Banach theorem
  In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
  
  
• Dual space
  Dual space
  In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
  
  
• Predual
  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. The predual of the space of differential forms is the space of chainlets....
  
  
• Weak topology
  Weak topology
  In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
  
  
• Reflexive space
  Reflexive space
  In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
  
  
• Polynomially reflexive space
  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....
  
  
• Baire category theorem
  Baire category theorem
  The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
  
  
• Open mapping theorem (functional analysis)
  Open mapping theorem (functional analysis)
  In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map...
  
  
• Closed graph theorem
  Closed graph theorem
  In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.- The closed graph theorem :...
  
  
• Uniform boundedness principle
  Uniform boundedness principle
  In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field...
  
  
• Arzelà–Ascoli theorem
• Banach–Alaoglu theorem
• Measure of non-compactness
  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.The underlying idea is the...
  
  
• Banach–Mazur theorem

Operator theory

Operator theory

In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....

• Bounded linear operator
  • Continuous linear extension
  • Compact operator
    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 of Y...
    
    
  • Approximation property
    Approximation property
    In mathematics, a Banach space is said to have the approximation property , if every compact operator is a limit of finite-rank operators. The converse is always true....
    
    
  • Invariant subspace
    Invariant subspace
    In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
    
    
• Spectral theory
  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. It is a result of studies of linear algebra and the solutions of...
  
  
  • Spectrum of an operator
  • Essential spectrum
    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".-The essential spectrum of self-adjoint operators:...
    
    
  • Spectral density
    Spectral density
    In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
    
    
• Topologies on the set of operators on a Hilbert space
  • norm topology
  • ultrastrong topology
  • strong operator topology
    Strong operator topology
    In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
    
    
  • weak operator topology
    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 is continuous for any vectors x and y in the Hilbert space.Equivalently, a...
    
    
  • weak-star operator topology
  • ultraweak topology
    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 of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B* of B, the trace class operators on H...
    
    
• S-number
  S-number
  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 nonnegative self-adjoint operator .The singular values are nonnegative real numbers, usually listed in...
  
  
• Fredholm operator
  Fredholm operator
  In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
  
  
• Fuglede's theorem
  Fuglede's theorem
  In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede.- The result :Theorem Let T and N be bounded operators on a complex Hilbert space with N being normal. If TN = NT, then TN* = N*T, where N* denotes the adjoint of N.Normality of N is necessary, as is seen by...
  
  
• Compression (functional analysis)
  Compression (functional analysis)
  In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operatorP_K T \vert_K : K \rightarrow K...
  
  
• Friedrichs extension
  Friedrichs extension
  In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician Kurt Friedrichs...
  
  
• Stone's theorem on one-parameter unitary groups
  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 which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators...
  
  
• Stone–von Neumann theorem
  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...
  
  
• Functional calculus
  Functional calculus
  In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. In mathematics, a functional calculus is a theory allowing one to apply mathematical...
  
  
  • Continuous functional calculus
    Continuous functional calculus
    In mathematics, the continuous functional calculus of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,Theorem...
    
    
  • Borel functional calculus
    Borel functional calculus
    In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...
    
    
• Hilbert–Pólya conjecture

Banach space examples

• Lp space
  Lp space
  In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
  
  
• Hardy space
  Hardy space
  In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
  
  
• Sobolev space
  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 as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
  
  
• Tsirelson space
  Tsirelson space
  In mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an l p space nor a c0 space can be embedded.It was introduced by B. S. Tsirelson in 1974...
  
  
• ba space
  Ba space
  In mathematics, the ba space ba of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive measures on \Sigma. The norm is defined as the variation, that is \|\nu\|=|\nu|....
  
  

Real and complex algebra

Associative algebra

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

s

• Uniform norm
• Matrix norm
• Spectral radius
  Spectral radius
  In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ.-Matrices:...
  
  
• Normed division algebra
  Normed division algebra
  In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....
  
  
• Stone–Weierstrass theorem
• Banach algebra
  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 which at the same time is also a Banach space...
  
  
• *-algebra
• B*-algebra
• C*-algebra
  • Universal C*-algebra
  • Spectrum of a C*-algebra
• Positive element
• Positive linear functional
  Positive linear functional
  In mathematics, especially in functional analysis, a positive linear functional on an ordered vector space is a linear functional f on V so that for all positive elements v of V, that is v≥0, it holds thatf\geq 0....
  
  
• operator algebra
  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...
  
  
  • nest algebra
    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. They were introduced by John Ringrose in the mid-1960s and have many interesting properties...
    
    
  • reflexive operator algebra
    Reflexive operator algebra
    In functional analysis, a reflexive operator algebra A is an operator algebra that has enough invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in A.This should not...
    
    
  • Calkin algebra
    Calkin algebra
    In functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of B, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K of compact operators....
    
    
• Gelfand representation
  Gelfand representation
  In mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
  
  
• Gelfand–Naimark theorem
  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
• Von Neumann algebra
  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. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
  
  
  • Abelian von Neumann algebra
    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.The prototypical example of an abelian von Neumann algebra is...
    
    
• von Neumann double commutant theorem
  • Commutant, bicommutant
• Topological ring
  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 mapswhere R × R carries the product topology.- General comments :...
  
  
• Noncommutative geometry
  Noncommutative geometry
  Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
  
  
• Disk algebra
  Disk algebra
  In function theory, the disk algebra A is the set of holomorphic functionswhere D is the open unit disk in the complex plane C, f extends to a continuous function on the closure of D...
  
  
• Colombeau algebra
  Colombeau algebra
  In mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions in which multiplication is not problematic...
  
  

Topological vector space

Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

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• Barrelled space
  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. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced,...
  
  
• Bornological space
  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...
  
  
• Bourbaki–Alaoglu theorem
• Dual pair
  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 form....
  
  
• F-space
  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...
  
  
• Fréchet space
  Fréchet space
  In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
  
  
• Krein–Milman theorem
• Locally convex topological vector space
  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 which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
  
  
• Mackey topology
  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...
  
  
• Mackey–Arens theorem
• Montel space
  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. Specifically, a Montel space is a barrelled topological vector space where every closed and bounded set is compact...
  
  
• Polar set
• Polar topology
• Seminorm

Wavelet

Wavelet

A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...

s

• Basis function
  Basis function
  In mathematics, a basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis...
  
  
• Daubechies wavelet
  Daubechies wavelet
  Named after Ingrid Daubechies, the Daubechies wavelets are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support...
  
  
• Haar wavelet
  Haar wavelet
  In mathematics, the Haar wavelet is a certain sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis...
  
  
• Morlet wavelet
• Mexican hat wavelet
• Complex mexican hat wavelet
  Complex mexican hat wavelet
  The complex Mexican hat wavelet is a low-oscillation, complex-valued, wavelet for the continuous wavelet transform. This wavelet is formulated in terms of its Fourier transformas the Hilbert analytic function of the conventional Mexican hat wavelet:...
  
  
• Hermitian wavelet
• Hermitian hat wavelet
  Hermitian hat wavelet
  The Hermitian hat wavelet is a low-oscillation, complex-valued wavelet.The real and imaginary parts of this wavelet are defined to be thesecond and first derivatives of a Gaussian respectively:...
  
  
• Discrete wavelet transform
  Discrete wavelet transform
  In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...
  
  
• Continuous wavelet
  Continuous wavelet
  In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either of time or of frequency....
  
  
• Continuous wavelet transform
  Continuous wavelet transform
  A continuous wavelet transform is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization...
  
  

Quantum theory

• Mathematical formulation of quantum mechanics
  Mathematical formulation of quantum mechanics
  The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
  
  
• Observable
  Observable
  In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
  
  
• Operator (physics)
  Operator (physics)
  In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
  
  
• Quantum state
  • Pure state
  • Fock state
    Fock state
    A Fock state , in quantum mechanics, is any element of a Fock space with a well-defined number of particles . These states are named after the Soviet physicist, V. A. Fock.-Definition:...
    
    , Fock space
    Fock space
    The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
    
    
  • Density state
  • Coherent state
    Coherent state
    In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resembles the oscillating behaviour of a classical harmonic oscillator...
    
    
• Heisenberg picture
  Heisenberg picture
  In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
  
  
• Density matrix
  Density matrix
  In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
  
  
• Quantum logic
  Quantum logic
  In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
  
  
• Quantum operation
  Quantum operation
  In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan...
  
  
• Wightman axioms
  Wightman axioms
  In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.The axioms exist in...
  
  

History

• Stefan Banach
  Stefan Banach
  Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
  
   (1892–1945)
• Hugo Steinhaus
  Hugo Steinhaus
  Władysław Hugo Dionizy Steinhaus was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics...
  
   (1887–1972)
• John von Neumann
  John von Neumann
  John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
  
  
• Alain Connes
  Alain Connes
  Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
  
   (born 1947)
• Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
• Earliest Known Uses of Some of the Words of Mathematics: Matrices and Linear Algebra