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- 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.".
- Hilbert_space thumbnail Standing_waves_on_a_string.gif?width=300.
- Hilbert_space wikiPageExternalLink books?as_isbn=0-486-65656-X.
- Hilbert_space wikiPageExternalLink books?as_isbn=0-8218-4790-2.
- Hilbert_space wikiPageExternalLink HilbertSpace.html.
- Hilbert_space wikiPageExternalLink 254a-notes-5-hilbert-spaces.
- Hilbert_space wikiPageID "20598932".
- Hilbert_space wikiPageRevisionID "603891063".
- Hilbert_space author "B.M. Levitan".
- Hilbert_space b "0".
- Hilbert_space b "w".
- Hilbert_space class "HistTopics".
- Hilbert_space date "1996".
- Hilbert_space hasPhotoCollection Hilbert_space.
- Hilbert_space id "Abstract_linear_spaces".
- Hilbert_space id "H/h047380".
- Hilbert_space id "p/h047380".
- Hilbert_space p "1".
- Hilbert_space p "2".
- Hilbert_space title "Abstract linear spaces".
- Hilbert_space title "Hilbert space".
- Hilbert_space subject Category:Concepts_in_physics.
- Hilbert_space subject Category:Hilbert_space.
- Hilbert_space subject Category:Linear_algebra.
- Hilbert_space subject Category:Operator_theory.
- Hilbert_space subject Category:Quantum_mechanics.
- Hilbert_space comment "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.".
- Hilbert_space label "Espace de Hilbert".
- Hilbert_space label "Espacio de Hilbert".
- Hilbert_space label "Espaço de Hilbert".
- Hilbert_space label "Hilbert space".
- Hilbert_space label "Hilbertraum".
- Hilbert_space label "Hilbertruimte".
- Hilbert_space label "Przestrzeń Hilberta".
- Hilbert_space label "Spazio di Hilbert".
- Hilbert_space label "Гильбертово пространство".
- Hilbert_space label "فضاء هيلبرت".
- Hilbert_space label "ヒルベルト空間".
- Hilbert_space label "希尔伯特空间".
- Hilbert_space sameAs Hilbertův_prostor.
- Hilbert_space sameAs Hilbertraum.
- Hilbert_space sameAs Χώρος_Χίλμπερτ.
- Hilbert_space sameAs Espacio_de_Hilbert.
- Hilbert_space sameAs Espace_de_Hilbert.
- Hilbert_space sameAs Spazio_di_Hilbert.
- Hilbert_space sameAs ヒルベルト空間.
- Hilbert_space sameAs 힐베르트_공간.
- Hilbert_space sameAs Hilbertruimte.
- Hilbert_space sameAs Przestrzeń_Hilberta.
- Hilbert_space sameAs Espaço_de_Hilbert.
- Hilbert_space sameAs m.03jrb.
- Hilbert_space sameAs Q190056.
- Hilbert_space sameAs Q190056.
- Hilbert_space wasDerivedFrom Hilbert_space?oldid=603891063.
- Hilbert_space depiction Standing_waves_on_a_string.gif.
- Hilbert_space isPrimaryTopicOf Hilbert_space.