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
79 pages build on this one.
- Rigged Hilbert space
- Spectral theorem for unbounded self-adjoint operators
- Subnormal operator
- Von Neumann's theorem on self-adjoint extensions
- Jones index theorem
- Toeplitz algebra
- Commutant lifting theorem
- Partial isometry
- Feldman–Hájek theorem
- Invariant subspace problem
- Tensor product of Hilbert spaces
- Calkin algebra
- Variational inequality
- von Neumann's inequality
- Toeplitz–Hausdorff theorem
- Coercive bilinear form
- Hilbert C*-module
- Proximal operator
- Spectral theorem for compact self-adjoint operators
- Ideal of compact operators
- Atkinson's theorem
- Algebra of bounded operators on a Hilbert space
- Von Neumann bicommutant theorem
- Ultraweak topology
- Trace class operator
- Kadison–Singer problem
- Geometric quantization
- Hyponormal operator
- Deficiency indices
- Hilbert sum
- Hilbert space adjoint
- Unitary operator
- Stone's theorem on one-parameter unitary groups
- Hilbert–Pólya conjecture
- Schatten class
- Weak operator topology
- Von Neumann algebra
- Mathematical formulation of quantum mechanics
- Hilbert projection theorem
- Hilbert dimension
- Lie–Trotter product formula
- Symmetric operator
- Borel functional calculus
- Nest algebra
- Adjoint of an unbounded operator
- Douglas' lemma
- Crossed product
- Grothendieck inequality
- Real spectrum of self-adjoint operators
- Hyperfinite type II1 factor
- Friedrichs extension
- Positive operator
- Reproducing kernel Hilbert space
- Murray–von Neumann classification of factors
- Lidskii's theorem
- Operator theory
- Projection-valued measure
- Tomita–Takesaki theory
- Operator space
- Weyl–von Neumann theorem
- Krein space
- Kaplansky density theorem
- Spectral triple
- Riesz representation theorem
- Fuglede's theorem
- Singular values of a compact operator
- Strong operator topology
- Spectral theorem for bounded normal operators
- Stinespring dilation theorem
- Weyl's theorem on compact perturbations
- Square root of a positive operator
- Numerical range
- Sz.-Nagy's dilation theorem
- Reflexive operator algebra
- Commutant
- Connes embedding problem
- Abstract Wiener space
- Essentially self-adjoint operator
- 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 | |
| Changed | 2 November 2026 |
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What links here
Used by
- Rigged Hilbert space
- Spectral theorem for unbounded self-adjoint operators
- Subnormal operator
- Von Neumann's theorem on self-adjoint extensions
- Commutant lifting theorem
- Partial isometry
- Feldman–Hájek theorem
- Invariant subspace problem
- Tensor product of Hilbert spaces
- Calkin algebra
- Variational inequality
- von Neumann's inequality
- Toeplitz–Hausdorff theorem
- Coercive bilinear form
- Hilbert C*-module
- Proximal operator
- Spectral theorem for compact self-adjoint operators
- Ideal of compact operators
- Algebra of bounded operators on a Hilbert space
- Von Neumann bicommutant theorem
- Ultraweak topology
- Trace class operator
- Kadison–Singer problem
- Geometric quantization
- Hyponormal operator
- Deficiency indices
- Hilbert sum
- Hilbert space adjoint
- Unitary operator
- Stone's theorem on one-parameter unitary groups
- Hilbert–Pólya conjecture
- Schatten class
- Weak operator topology
- Von Neumann algebra
- Mathematical formulation of quantum mechanics
- Hilbert projection theorem
- Hilbert dimension
- Lie–Trotter product formula
- Symmetric operator
- Borel functional calculus
- Nest algebra
- Adjoint of an unbounded operator
- Douglas' lemma
- Crossed product
- Grothendieck inequality
- Real spectrum of self-adjoint operators
- Hyperfinite type II1 factor
- Friedrichs extension
- Positive operator
- Reproducing kernel Hilbert space
- Murray–von Neumann classification of factors
- Lidskii's theorem
- Operator theory
- Projection-valued measure
- Tomita–Takesaki theory
- Operator space
- Weyl–von Neumann theorem
- Krein space
- Kaplansky density theorem
- Spectral triple
- Riesz representation theorem
- Fuglede's theorem
- Singular values of a compact operator
- Strong operator topology
- Spectral theorem for bounded normal operators
- Stinespring dilation theorem
- Weyl's theorem on compact perturbations
- Square root of a positive operator
- Numerical range
- Sz.-Nagy's dilation theorem
- Reflexive operator algebra
- Commutant
- Abstract Wiener space
- Essentially self-adjoint operator
- Hellinger–Toeplitz theorem
Instances
Special cases
Results about this
- von Neumann's inequality
- Hilbert projection theorem
- Douglas' lemma
- Grothendieck inequality
- Riesz representation theorem