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Hilbert space
Definition
A real Hilbert space is a real inner-product space (Real and complex inner-product spaces and their induced length) whose induced-length metric is complete in the sense of Complete metric space: every Cauchy sequence converges in the space: every Cauchy sequence in for the norm (The induced length is a norm) converges to a point of . A complex Hilbert space is a complex inner-product space with the same completeness property, so that a Hilbert space is exactly a real or complex inner-product space that is a Banach space for its induced norm (Banach space).
The completion convention is the Cauchy-sequence one. Banach space defines completeness by convergence of Cauchy sequences, and this page keeps that convention throughout. It is weaker than -completeness — the assertion that every decreasing sequence of nonempty closed subsets with diameters tending to zero has a nonempty intersection. The two agree in ZFC, but Blackadar, Farah and Karagila note that the closest-point theorem on a -complete inner-product space is provable in ZF, whereas the Cauchy-complete form used below consumes the Axiom of Countable Choice; nothing here silently imports the stronger notion.
Depends on
Used by
- A separable infinite-dimensional Hilbert space is ℓ² Corollary
- Compact operator iff approximation numbers tend to zero Corollary
- Finite rank operators are norm dense in compact Hilbert space operators Corollary
- Orthonormal eigenbasis for a compact self adjoint operator Corollary
- An everywhere-defined closed operator on a Banach space is bounded Counterexample
- An inner-product space need not be complete Counterexample
- Nearest-point maps to convex sets need not be linear Counterexample
- Absolute value and singular values of a compact operator Definition
- Adjoint of a densely defined operator Definition
- Cyclic vector and cyclic normal operator Definition
- Densely defined, closed and closable operators, and cores Definition
- Hilbert–Schmidt operator and Hilbert–Schmidt norm Definition
- Infinitesimal generator of a unitary group Definition
- Integral of a measurable function against a projection-valued measure Definition
- Isometry coisometry and partial isometry Definition
- Numerical range and numerical radius Definition
- Order on bounded self adjoint operators Definition
- Orthonormal families, complete orthonormal systems and Hilbert bases Definition
- Projection valued measure Definition
- Resolvent and spectrum of an unbounded operator Definition
- Spectral multiplicity function in the separable case Definition
- Strongly continuous one-parameter unitary group Definition
- Trace class operator Definition
- Trace of a trace class operator Definition
- Unbounded linear operators: domain, graph and extension Definition
- Adjoint, norm and trace of an operator of rank at most one Example
- Borel functional calculus defines a discontinuous characteristic function Example
- Diagonal Schatten class criteria on ell two Example
- Functional calculus for a diagonal operator Example
- Integral operator trace under a valid diagonal hypothesis Example
- Pvm of a diagonal normal operator Example
- Pvm of a multiplication operator Example
- Spectral projection of an isolated eigenvalue agrees with the riesz projection Example
- The standard inner products make K n, ell two and quotient L two Hilbert spaces Example
- Volterra operator is Hilbert Schmidt and quasinilpotent Example
- Bounded Hilbert operators form a C star algebra Lemma
- Continuous functional calculus produces a regular PVM Lemma
- Eigenspaces of a self adjoint operator are orthogonal Lemma
- L² with the integral pairing is a Hilbert space Lemma
- Maximal orthogonal family of cyclic reducing subspaces Lemma
…and 39 more results.
Dependency tree · two levels
20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 1.41, p.39 (standard reference, not scraped)
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lecture 16 (standard reference, not scraped)
- Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, Definitions 1.0.1 and 2.0.1 (standard reference, not scraped)