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Type of vector space in math
The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It extends the methods of Euclidean geometry and calculus from
Hilbert_space
Space-filling curve
The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician
Hilbert_curve
Construction for adding objects to a Hilbert space
rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction which can enlarge a Hilbert space to a bigger space containing
Rigged_Hilbert_space
Vector space with generalized dot product
product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If
Inner_product_space
Fundamental space of geometry
point. Mathematics portal Hilbert space, a generalization to infinite dimension, used in functional analysis Position space, an application in physics
Euclidean_space
Tensor product space endowed with a special inner product
product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
In functional analysis, a Hilbert space
kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H {\displaystyle
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
German mathematician (1862–1943)
Hilbert ring Hilbert–Poincaré series Hilbert series and Hilbert polynomial Hilbert space Hilbert spectrum Hilbert system Hilbert transform Hilbert's arithmetic
David_Hilbert
On surjectivity of linear map to anti-dual
and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessary and sufficient condition for a Hausdorff pre-Hilbert space to be
Fundamental theorem of Hilbert spaces
Fundamental_theorem_of_Hilbert_spaces
Mathematical description of quantum state
multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product of two wave functions is a measure of the overlap
Wave_function
Topic in mathematics
{\displaystyle A\colon H\to H} that acts on a Hilbert space H {\displaystyle H} and has finite Hilbert–Schmidt norm ‖ A ‖ HS 2 = def ∑ i ∈ I ‖ A e
Hilbert–Schmidt_operator
Surjective bounded operator on a Hilbert space preserving the inner product
analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations
Unitary_operator
Functional analysis concept
compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Type of convergence in Hilbert spaces
a Hilbert space is the convergence of a sequence of points in the weak topology. A sequence of points ( x n ) {\displaystyle (x_{n})} in a Hilbert space
Weak convergence (Hilbert space)
Weak_convergence_(Hilbert_space)
Notation for quantum states
typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable
Bra–ket_notation
Multi particle state space
variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932
Fock_space
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite-dimensional Euclidean spaces. They were introduced
Nuclear_space
Integral transform and linear operator
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces
Hilbert_transform
Property of a mathematical space
Higher dimensions Vector space Plane of rotation Curse of dimensionality String theory Infinite Hilbert space Function space Dimension (data warehouse)
Dimension
Algebraic structure in linear algebra
of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. In this article, vectors
Vector_space
Mathematical concept
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
Semi-Hilbert_space
Generalized Euclidean space in mathematics
quantum mechanics, the projective Hilbert space or ray space P ( H ) {\displaystyle \mathbf {P} (H)} of a complex Hilbert space H {\displaystyle H} is the set
Projective_Hilbert_space
Vector space in functional analysis
{\displaystyle J^{3}=J.\,} The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner
Indefinite inner product space
Indefinite_inner_product_space
Operation in abstract algebra
integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module
Direct_sum_of_modules
Mathematical set with some added structure
topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.[better source needed] A space consists of
Space_(mathematics)
Normed vector space that is complete
"Banach space" and Banach in turn then coined the term "Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet
Banach_space
Classification of irreducible representations of the Poincaré group
representations of the Poincaré group. After all, two vectors in the quantum Hilbert space that differ by multiplication by a constant represent the same physical
Wigner's_classification
*-algebra of bounded operators on a Hilbert space
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
Von_Neumann_algebra
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such
Kuiper's_theorem
Formulation of classical mechanics in terms of Hilbert spaces
mechanics, based on a Hilbert space of complex, square-integrable functions representing classical observables on phase spaces. As its name suggests,
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Generalized function whose value is zero everywhere except at zero
the delta function in this Hilbert space. A Hilbert space having such a kernel is called a reproducing kernel Hilbert space. In the special case of the
Dirac_delta_function
In mathematics, vector space of linear forms
spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early
Dual_space
Type of continuous linear operator
eigenvalues, to compact normal operators on a complex Hilbert space. A general compact operator on a Hilbert space need not be self-adjoint or normal. Nevertheless
Compact_operator
Conjugate transpose of an operator in infinite dimensions
an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H {\displaystyle H} . The definition has been further extended to include
Hermitian_adjoint
Formulation of quantum mechanics on a Hilbert Space
in terms of operators on a Hilbert space. They were introduced by Paul Dirac in 1930 and John von Neumann in 1932. The space H {\displaystyle \mathbb {H}
Dirac–von_Neumann_axioms
Curve whose range contains the unit square
analytic form of the Hilbert curve, however, is more complicated than Peano's. Let C {\displaystyle {\mathcal {C}}} denote the Cantor space 2 N {\displaystyle
Space-filling_curve
Linear operator in functional analysis
Exactly the same argument shows that an operator T {\displaystyle T} on a Hilbert space H {\displaystyle H} is of rank 1 {\displaystyle 1} if and only if T
Finite-rank_operator
Theorem in the mathematical formulation of quantum mechanics
represented on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor
Wigner's_theorem
Type of topological space
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore
Hilbert_cube
Mathematical entity to describe the probability of each possible measurement on a system
vector in a Hilbert space. Mixed states are statistical mixtures of pure states and cannot be represented as vectors on that Hilbert space, and instead
Quantum_state
Axiomatization of probability and physics
Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent. It occurs on the widely cited list of Hilbert's
Hilbert's_sixth_problem
In mathematics, a linear operator acting on inner product space
below that for a complex Hilbert space the self adjointness follows automatically from non-negativity. For a real Hilbert space non-negativity does not
Positive_operator
Mathematical concept
topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the
Weak_topology
Mathematical construction relating to infinite-dimensional spaces
can be represented by the abstract Wiener space construction. Let H {\displaystyle H} be a real Hilbert space, assumed to be infinite dimensional and separable
Abstract_Wiener_space
Manifold modelled on Hilbert spaces
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood
Hilbert_manifold
functioning under proper operator behavior. T on a Hilbert space H is an operator on a larger Hilbert space K, whose restriction to H composed with the orthogonal
Dilation_(operator_theory)
showed Traveling Salesman Theorem also holds for sets E that lie in any Hilbert Space, and in particular, implies the theorems of Jones and Okikiolu, where
Analyst's traveling salesman theorem
Analyst's_traveling_salesman_theorem
Concepts in convex analysis
Alternatively, many authors define the dual cone in the context of a real Hilbert space (such as Rn equipped with the Euclidean inner product) to be what is
Dual_cone_and_polar_cone
On closed convex subsets in Hilbert space
mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x {\displaystyle x} in a Hilbert space H {\displaystyle
Hilbert_projection_theorem
Area of mathematics
complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm
Functional_analysis
R-tree variant and index for multidimensional objects
clusters the data rectangles on a node. Hilbert R-trees use space-filling curves, and specifically the Hilbert curve, to impose a linear ordering on the
Hilbert_R-tree
Description of physical properties at the atomic and subatomic scale
points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the
Quantum_mechanics
Bounded operators with sub-unit norm
contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias. If T is a contraction acting on a Hilbert space H {\displaystyle
Contraction_(operator_theory)
Description of a quantum-mechanical system
nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of square-integrable
Schrödinger_equation
Result about when a matrix can be diagonalized
on Hilbert spaces. The spectral theorem also provides a canonical decomposition, called the spectral decomposition, of the underlying vector space on
Spectral_theorem
Regularization technique for ill-posed problems
above we can interpret A {\displaystyle A} as a compact operator on Hilbert spaces, and x {\displaystyle x} and b {\displaystyle b} as elements in the
Ridge_regression
Function spaces generalizing finite-dimensional p norm spaces
the space of square-summable sequences, which is a Hilbert space, and ℓ ∞ , {\displaystyle \ell ^{\infty },} the space of bounded sequences. The space of
Lp_space
Model of interacting spinless bosons on a lattice
for example). In the case of a mixture, the Hilbert space is simply the tensor product of the Hilbert spaces of the individual species. Typically additional
Bose–Hubbard_model
Generalization of the concept of a direct sum in mathematics
integral or Hilbert integral is a generalization of the concept of a direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct
Direct_integral
Measure of the "size" of linear operators
the sequence space ℓ ∞ {\displaystyle \ell ^{\infty }} is not separable. The associative algebra of all bounded operators on a Hilbert space, together with
Operator_norm
Topological complex vector space
of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the
C*-algebra
Mathematical theorem
the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used in the reproducing kernel Hilbert space theory
Mercer's_theorem
Topics referred to by the same term
Hilbert operator may refer to: The epsilon operator in Hilbert's epsilon calculus The Hilbert–Schmidt operators on a Hilbert space Hilbert–Schmidt integral
Hilbert_operator
Mathematics concept
{\overline {\overline {V}}}} is identical to V . {\displaystyle V.} Given a Hilbert space H {\displaystyle {\mathcal {H}}} (either finite or infinite dimensional)
Complex conjugate of a vector space
Complex_conjugate_of_a_vector_space
Topologies on operators on a Hilbert space
the arrows pointing from strong to weak. If H is a Hilbert space, the linear space of Hilbert space operators B(X) has a (unique) predual B ( H ) ∗ {\displaystyle
Operator_topologies
Hungarian and American mathematician and physicist (1903–1957)
the Hilbert space associated with the quantum system. The physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear
John_von_Neumann
Gauge field loop operator
charged under the gauge group. Its charge forms a quantized internal Hilbert space, which can be integrated out, yielding the Wilson line as the world-line
Wilson_loop
the Hilbert space ( L 2 ) := L 2 ( S ′ ( R ) , μ ) , {\displaystyle (L^{2}):=L^{2}\left(S'(\mathbb {R} ),\mu \right),} generalizing the Hilbert spaces L
White_noise_analysis
Matrix decomposition
{\displaystyle \mathbf {M} } on a (possibly infinite-dimensional) Hilbert space, that ‖ M ‖ op = ‖ M ∗ M ‖ op . {\displaystyle \|\mathbf {M} \|_{\text{op}}={\sqrt
Singular_value_decomposition
Theorem in quantum mechanics
each physical system is associated with a Hilbert space. For the purposes of this overview, the Hilbert space is assumed to be finite-dimensional. In the
Gleason's_theorem
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
Felix Klein, the first edition incorporated Hilbert's treatment of the Dirichlet problem using Hilbert space techniques; Brouwer's contributions to topology;
Uniformization_theorem
Theory of logic to account for observations from quantum theory
separable Hilbert space, Constantin Piron, Günther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space. Inspired
Quantum_logic
Space of stochastic processes
of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to C 0 . {\displaystyle C_{0}.} Classical Wiener measure
Classical_Wiener_space
Mathematical structures that allow quantum mechanics to be explained
uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Zeta-like functions approximate arbitrary holomorphic functions
the Bergman space, falsely named Hardy space in (Voronin and Karatsuba, 1992), in H of holomorphic functions defined on U, a Hilbert space. We set u k
Zeta_function_universality
Theorem in physics
orthonormal bases for a Hilbert space represent measurements that can be performed upon a system having that Hilbert space. Each vector in a basis represents
Bell's_theorem
particular Hilbert space. The construction below proceeds first by defining operators that correspond to the Heisenberg group generators. Next, the Hilbert space
Theta_representation
Method for solving continuous operator problems (such as differential equations)
Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space V {\displaystyle V} , namely, find u ∈ V {\displaystyle u\in V} such
Galerkin_method
Concept in linear algebra
C\right\}.} If C {\displaystyle C} is a closed vector subspace of a Hilbert space H {\displaystyle H} then H = C ⊕ C ⊥ and ( C ⊥ ) ⊥ = C {\displaystyle
Orthogonal_complement
methods typically involves reproducing kernel Hilbert spaces (RKHS). Not all kernels form inner product spaces, as they may not always be positive semidefinite
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Any entity that can be measured
linear self-adjoint operators on a separable complex Hilbert space representing the quantum state space. Observables assign values to outcomes of particular
Observable
dimension. In Hilbert spaces such operators are usually called trace class operators and one can define such things as the trace. In Banach spaces this is no
Nuclear operators between Banach spaces
Nuclear_operators_between_Banach_spaces
Recipe for constructing a quantum analog of a classical physical theory
self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the
Geometric_quantization
Space of all possible states that a system can take
mechanics, the coordinates p and q of phase space normally become Hermitian operators in a Hilbert space. But they may alternatively retain their classical
Phase_space
Fundamental theorem in condensed matter physics
eigenstate decompositions in a Hilbert space are in some sense purely formal: The decomposition series do not converge in Hilbert space, and no proper spatially
Bloch's_theorem
Construction in functional analysis, useful to solve differential equations
reflexivity no longer holds. Hilbert spaces are Banach spaces, so the above discussion applies to bounded operators on Hilbert spaces as well. A subtle point
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
the Malliavin calculus, a Gaussian probability space is a probability space together with a Hilbert space of mean zero, real-valued Gaussian random variables
Gaussian_probability_space
Representation of a quantum mechanical system
each quantum mechanical system is associated with a separable complex Hilbert space H {\displaystyle H} . A pure state of a quantum system is represented
Bloch_sphere
Liouville space, also known as line space, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Schmidt
Liouville_space
Mathematical space with a notion of closeness
found in general topology Exterior space Hausdorff space – Type of topological space Hilbert space – Type of vector space in math Hemicontinuity – Semicontinuity
Topological_space
Topological space that is connected
removal of countably many points. Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field (such as R {\displaystyle \mathbb
Connected_space
Function valued in a vector space; typically a real or complex one
componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space. Most of the above hold
Vector-valued_function
Representation of the symmetry group of spacetime in special relativity
collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Collection of mathematical theories
The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms
Spectral_theory
Process in quantum mechanical theories
quantum state. Observables are represented by operators acting on a Hilbert space of such quantum states. The eigenvalue of an operator acting on one
Canonical_quantization
Decomposition of periodic functions
context of Hilbert spaces. For example, the space of square-integrable functions on [ − π , π ] {\displaystyle [-\pi ,\pi ]} forms the Hilbert space L 2 (
Fourier_series
Non-linear generalization of a Hilbert space
In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently
Hadamard_space
Generalized measurement in quantum mechanics
is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and
POVM
Hilbert space of square-integrable holomorphic functions of n complex variables
Lebesgue measure on C n . {\displaystyle \mathbb {C} ^{n}.} It is a Hilbert space with respect to the associated inner product: ⟨ F ∣ G ⟩ = π − n ∫ C
Segal–Bargmann_space
Banach space are a classification of Banach spaces through probability theory and a measure for how far a Banach space is away from being a Hilbert space. The
Type and cotype of a Banach space
Type_and_cotype_of_a_Banach_space
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