Matches in DBpedia 2016-04 for { <http://dbpedia.org/resource/Hilbert_space> ?p ?o }
- Hilbert_space abstract "The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of \"dropping the altitude\" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.".
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- Hilbert_space wikiPageExternalLink book-schroe.
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- Hilbert_space wikiPageExternalLink 254a-notes-5-hilbert-spaces.
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- Hilbert_space wikiPageWikiLink Absolute_convergence.
- Hilbert_space wikiPageWikiLink Absolute_value.
- Hilbert_space wikiPageWikiLink Altitude_(triangle).
- Hilbert_space wikiPageWikiLink American_Mathematical_Monthly.
- Hilbert_space wikiPageWikiLink American_Mathematical_Society.
- Hilbert_space wikiPageWikiLink Antilinear_map.
- Hilbert_space wikiPageWikiLink Applied_mathematics.
- Hilbert_space wikiPageWikiLink Atiyah–Singer_index_theorem.
- Hilbert_space wikiPageWikiLink Banach_algebra.
- Hilbert_space wikiPageWikiLink Banach_space.
- Hilbert_space wikiPageWikiLink Banach–Alaoglu_theorem.
- Hilbert_space wikiPageWikiLink Basis_(linear_algebra).
- Hilbert_space wikiPageWikiLink Bergman_kernel.
- Hilbert_space wikiPageWikiLink Bergman_space.
- Hilbert_space wikiPageWikiLink Bessel_potential.
- Hilbert_space wikiPageWikiLink Bijection.
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- Hilbert_space wikiPageWikiLink Bolzano–Weierstrass_theorem.
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- Hilbert_space wikiPageWikiLink C*-algebra.
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- Hilbert_space wikiPageWikiLink Category:Hilbert_space.
- Hilbert_space wikiPageWikiLink Category:Linear_algebra.
- Hilbert_space wikiPageWikiLink Category:Operator_theory.
- Hilbert_space wikiPageWikiLink Category:Quantum_mechanics.
- Hilbert_space wikiPageWikiLink Cauchy_sequence.
- Hilbert_space wikiPageWikiLink Cauchys_convergence_test.
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- Hilbert_space wikiPageWikiLink Cauchy–Schwarz_inequality.
- Hilbert_space wikiPageWikiLink Change_of_basis.
- Hilbert_space wikiPageWikiLink Chaos_theory.
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- Hilbert_space wikiPageWikiLink Closure_(topology).
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- Hilbert_space wikiPageWikiLink Cokernel.
- Hilbert_space wikiPageWikiLink Compact_convergence.
- Hilbert_space wikiPageWikiLink Compact_group.
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- Hilbert_space wikiPageWikiLink Complex_analysis.
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- Hilbert_space wikiPageWikiLink Countable_set.
- Hilbert_space wikiPageWikiLink David_Hilbert.
- Hilbert_space wikiPageWikiLink De_Rham_cohomology.
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- Hilbert_space wikiPageWikiLink Degrees_of_freedom_(mechanics).
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- Hilbert_space wikiPageWikiLink Density_matrix.
- Hilbert_space wikiPageWikiLink Derivative.
- Hilbert_space wikiPageWikiLink Differential_geometry.
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- Hilbert_space wikiPageWikiLink Direct_method_in_the_calculus_of_variations.
- Hilbert_space wikiPageWikiLink Direct_sum_of_modules.
- Hilbert_space wikiPageWikiLink Dirichlet_boundary_condition.
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- Hilbert_space wikiPageWikiLink Dot_product.
- Hilbert_space wikiPageWikiLink Dual_space.
- Hilbert_space wikiPageWikiLink Dynamical_system.
- Hilbert_space wikiPageWikiLink Eberlein–Šmulian_theorem.
- Hilbert_space wikiPageWikiLink Eigenfunction.
- Hilbert_space wikiPageWikiLink Eigenvalues_and_eigenvectors.
- Hilbert_space wikiPageWikiLink Elliptic_partial_differential_equation.
- Hilbert_space wikiPageWikiLink Ergodic_theory.
- Hilbert_space wikiPageWikiLink Erhard_Schmidt.
- Hilbert_space wikiPageWikiLink Ernst_Sigismund_Fischer.
- Hilbert_space wikiPageWikiLink Euclidean_space.