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Statements (105)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:Vector
gptkb:mathematical_concept gptkb:normed_vector_space gptkb:topology |
| gptkbp:alternativeName |
Banach_space
infinite-dimensional_Hilbert_space |
| gptkbp:basisFor |
orthonormal basis
uncountable or countable orthonormal basis |
| gptkbp:containsElement |
gptkb:Vector
|
| gptkbp:contrastsWith |
gptkb:finite-dimensional_Hilbert_space
|
| gptkbp:defines |
A normed vector space that is complete with respect to the norm topology.
|
| gptkbp:dimensions |
finite or infinite
infinite |
| gptkbp:example |
gptkb:L2_space
gptkb:finite-dimensional_Euclidean_space gptkb:L^p_space gptkb:l^p_space gptkb:sequence_space_l2 l2 space C([0,1]) sequence space space of continuous functions |
| gptkbp:fieldOfStudy |
gptkb:mathematics
functional analysis |
| gptkbp:generalizes |
gptkb:Euclidean_space
gptkb:Hilbert_space |
| gptkbp:hasApplication |
gptkb:machine_learning
gptkb:probability_theory Fourier analysis control theory partial differential equations statistics |
| gptkbp:hasAxiom |
gptkb:Cauchy-Schwarz_inequality
gptkb:Pythagorean_theorem_(generalized) gptkb:parallelogram_law |
| gptkbp:hasDualSpace |
gptkb:Hilbert_space
itself (Riesz representation theorem) |
| gptkbp:hasMorphism |
bounded linear operator
|
| gptkbp:hasNorm |
induced by inner product
|
| gptkbp:hasProperty |
gptkb:spectral_theorem
bounded linear operators compact operators completeness direct sum decomposition inner product orthogonal complement orthogonality orthonormal sequence projection theorem self-adjoint operators separability (in some cases) unitary operators complete normed complete inner product space |
| gptkbp:hasQuotientSpace |
gptkb:Hilbert_space
|
| gptkbp:hasSpecialCase |
gptkb:Hilbert_space
|
| gptkbp:hasSubgroup |
gptkb:Hilbert_space
|
| gptkbp:hasTopology |
norm topology
|
| https://www.w3.org/2000/01/rdf-schema#label |
Hilbert space
|
| gptkbp:introducedIn |
1920s
|
| gptkbp:isCompleteWithRespectTo |
gptkb:standard
|
| gptkbp:namedAfter |
gptkb:David_Hilbert
gptkb:Stefan_Banach |
| gptkbp:operator |
inner product
scalar multiplication vector addition |
| gptkbp:structure |
gptkb:normed_vector_space
gptkb:topology inner product space linear space topological vector space |
| gptkbp:subclassOf |
gptkb:Hilbert_space
|
| gptkbp:used_in |
functional analysis
mathematical physics operator theory quantum mechanics |
| gptkbp:usedIn |
gptkb:machine_learning
gptkb:probability_theory gptkb:signal_processing gptkb:C*-algebra Fourier analysis control theory data science ergodic theory functional analysis harmonic analysis mathematical physics measure theory numerical analysis operator theory partial differential equations quantum mechanics statistics stochastic processes approximation theory optimization spectral theory convex analysis distribution theory fixed point theory functional equations topological vector spaces |
| gptkbp:bfsParent |
gptkb:David_Hilbert
|
| gptkbp:bfsLayer |
3
|