Functional analysis Hilbert spaces Concept
Hilbert space
Summary
A Hilbert space is an inner product space that is complete for the norm induced by its inner product. So it is a Banach space whose geometry has angles and orthogonality, as in Euclidean space. Every Hilbert space has an orthonormal basis, and every closed subspace has an orthogonal complement. Every continuous linear functional is the inner product with a fixed vector. Hilbert spaces are the setting of Fourier series, of spectral theory and of quantum mechanics.
Where this leads
158 pages build on this one.
- Rigged Hilbert space
- Spectral theorem for unbounded self-adjoint operators
- Resistance form
- Functional data analysis
- Ergodic decomposition
- Subnormal operator
- Von Neumann's theorem on self-adjoint extensions
- Generalized Fourier series
- Wold's decomposition theorem
- Jones index theorem
- Hilbert–Schmidt expansion theorem
- L2 convergence of Fourier series
- Oscillator representation
- Toeplitz algebra
- Commutant lifting theorem
- Partial isometry
- Feldman–Hájek theorem
- Céa's lemma
- Inertial manifold
- Induced representation of a locally compact group
- Representer theorem
- Invariant subspace problem
- Tensor product of Hilbert spaces
- Bilinear series of a symmetric kernel
- Calkin algebra
- Bargmann transform
- Variational inequality
- Peter–Weyl theorem
- Gelfand–Naimark–Segal construction
- von Neumann's inequality
- Kingman's subadditive ergodic theorem
- Lax–Milgram theorem
- Landweber iteration
- Determinant line bundle
- Toeplitz–Hausdorff theorem
- Schwarz alternating method
- Coercive bilinear form
- Hilbert C*-module
- Proximal operator
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Minimum total potential energy principle
- Ideal of compact operators
- Atkinson's theorem
- Rayleigh–Ritz method
- Algebra of bounded operators on a Hilbert space
- Nyquist–Shannon sampling theorem
- Von Neumann bicommutant theorem
- Inversion of the Weierstrass transform
- Ultraweak topology
- Trace class operator
- Kadison–Singer problem
- Entropic uncertainty principle
- Geometric quantization
- Hyponormal operator
- Finite element method
- Deficiency indices
- Mackey imprimitivity theorem
- Mixed finite element method
- Hilbert sum
- Hilbert space adjoint
- Plancherel theorem for groups
- Unitary operator
- Stiffness matrix
- Stone's theorem on one-parameter unitary groups
- Dirichlet form
- Unitary representation
- Admissible representation
- Hilbert–Pólya conjecture
- Haagerup property
- Schatten class
- Heisenberg inequality for the Fourier transform
- Weak operator topology
- Lidskii's trace formula for integral operators
- Von Neumann algebra
- Mathematical formulation of quantum mechanics
- Babenko–Beckner inequality
- Min-max theorem
- Finite element exterior calculus
- Spectrum of the Laplace–Beltrami operator
- Obstacle problem
- Lions–Stampacchia theorem
- Plancherel theorem
- Hausdorff–Young inequality
- Hilbert projection theorem
- Birkhoff's ergodic theorem
- Stefan problem as a variational inequality
- Hilbert dimension
- Open embedding theorem for Hilbert manifolds
- Strang splitting
- Lie–Trotter product formula
- Symmetric operator
- Borel functional calculus
- Picard's criterion for first-kind equations
- Galerkin method
- Nest algebra
- Adjoint of an unbounded operator
- Monotone operator method for Hammerstein equations
- Oseledets theorem
- Douglas' lemma
- Hilbert–Schmidt operator
- Crossed product
- Grothendieck inequality
- Fourier basis of L2 of the circle
- Real spectrum of self-adjoint operators
- Hyperfinite type II1 factor
- Friedrichs extension
- LOBPCG
- Positive operator
- Mercer's theorem
- Reproducing kernel Hilbert space
- von Neumann mean ergodic theorem
- Murray–von Neumann classification of factors
- Lidskii's theorem
- Operator theory
- Signorini problem
- Feichtinger conjecture
- Matching pursuit
- Projection-valued measure
- Tomita–Takesaki theory
- Petrov–Galerkin method
- Operator space
- Frame operator
- Weyl–von Neumann theorem
- Ladyzhenskaya–Babuška–Brezzi condition
- Fourier multiplier
- Krein space
- Tight frame
- Kaplansky density theorem
- Spectral triple
- Riesz representation theorem
- Quantum logic
- Basel problem via Fourier series
- Fuglede's theorem
- Singular values of a compact operator
- Plancherel theorem for abelian groups
- Riesz basis
- Pesin theory
- Strong operator topology
- Schmidt's singular value decomposition of a kernel
- Regularization of ill-posed problems
- Spectral theorem for bounded normal operators
- Stinespring dilation theorem
- Weyl's theorem on compact perturbations
- Cotlar–Stein lemma
- Square root of a positive operator
- Numerical range
- Sz.-Nagy's dilation theorem
- Reflexive operator algebra
- Commutant
- Connes embedding problem
- Whittaker–Shannon interpolation formula
- Abstract Wiener space
- Anderson–Kadec theorem
- Stone–von Neumann theorem
- Essentially self-adjoint operator
- Tikhonov regularization
- Hellinger–Toeplitz theorem
Further reading
About this page
| Tag | HM2HHE3J51 |
|---|---|
| URI | m2://HM2HHE3J51 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 46C05 |
| Specializes | Banach space, Inner product space |
| Same as |
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What links here
Used by
- Rigged Hilbert space
- Spectral theorem for unbounded self-adjoint operators
- Resistance form
- Functional data analysis
- Subnormal operator
- Von Neumann's theorem on self-adjoint extensions
- Generalized Fourier series
- Wold's decomposition theorem
- Commutant lifting theorem
- Partial isometry
- Feldman–Hájek theorem
- Céa's lemma
- Inertial manifold
- Induced representation of a locally compact group
- Representer theorem
- Invariant subspace problem
- Tensor product of Hilbert spaces
- Calkin algebra
- Bargmann transform
- Variational inequality
- Peter–Weyl theorem
- Gelfand–Naimark–Segal construction
- von Neumann's inequality
- Lax–Milgram theorem
- Landweber iteration
- Determinant line bundle
- Toeplitz–Hausdorff theorem
- Schwarz alternating method
- Coercive bilinear form
- Hilbert C*-module
- Proximal operator
- Spectral theorem for compact self-adjoint operators
- Ideal of compact operators
- Rayleigh–Ritz method
- Algebra of bounded operators on a Hilbert space
- Von Neumann bicommutant theorem
- Ultraweak topology
- Trace class operator
- Kadison–Singer problem
- Geometric quantization
- Hyponormal operator
- Finite element method
- Deficiency indices
- Mackey imprimitivity theorem
- Mixed finite element method
- Hilbert sum
- Hilbert space adjoint
- Plancherel theorem for groups
- Unitary operator
- Stiffness matrix
- Stone's theorem on one-parameter unitary groups
- Dirichlet form
- Unitary representation
- Admissible representation
- Hilbert–Pólya conjecture
- Haagerup property
- Schatten class
- Weak operator topology
- Von Neumann algebra
- Mathematical formulation of quantum mechanics
- Min-max theorem
- Finite element exterior calculus
- Lions–Stampacchia theorem
- Plancherel theorem
- Hilbert projection theorem
- Hilbert dimension
- Open embedding theorem for Hilbert manifolds
- Lie–Trotter product formula
- Symmetric operator
- Borel functional calculus
- Picard's criterion for first-kind equations
- Galerkin method
- Nest algebra
- Adjoint of an unbounded operator
- Monotone operator method for Hammerstein equations
- Douglas' lemma
- Hilbert–Schmidt operator
- Crossed product
- Grothendieck inequality
- Fourier basis of L2 of the circle
- Real spectrum of self-adjoint operators
- Hyperfinite type II1 factor
- Friedrichs extension
- Positive operator
- Reproducing kernel Hilbert space
- von Neumann mean ergodic theorem
- Murray–von Neumann classification of factors
- Lidskii's theorem
- Operator theory
- Feichtinger conjecture
- Matching pursuit
- Projection-valued measure
- Tomita–Takesaki theory
- Petrov–Galerkin method
- Operator space
- Frame operator
- Weyl–von Neumann theorem
- Ladyzhenskaya–Babuška–Brezzi condition
- Krein space
- Tight frame
- Kaplansky density theorem
- Spectral triple
- Riesz representation theorem
- Quantum logic
- Fuglede's theorem
- Singular values of a compact operator
- Plancherel theorem for abelian groups
- Riesz basis
- Strong operator topology
- Regularization of ill-posed problems
- Spectral theorem for bounded normal operators
- Stinespring dilation theorem
- Weyl's theorem on compact perturbations
- Cotlar–Stein lemma
- Square root of a positive operator
- Numerical range
- Sz.-Nagy's dilation theorem
- Reflexive operator algebra
- Commutant
- Abstract Wiener space
- Anderson–Kadec theorem
- Stone–von Neumann theorem
- Essentially self-adjoint operator
- Tikhonov regularization
- Hellinger–Toeplitz theorem
Instances
Special cases
Results about this
- von Neumann's inequality
- Hilbert projection theorem
- Douglas' lemma
- Grothendieck inequality
- Riesz representation theorem