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Hilbert space

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IP属地:安徽1楼2017-11-01 16:51回复
    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 the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. 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.


    IP属地:安徽2楼2017-11-01 16:52
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      目录 [隐藏]
      1Definition and illustration
      1.1Motivating example: Euclidean space
      1.2Definition
      1.3Second example: sequence spaces
      2History
      3Examples
      3.1Lebesgue spaces
      3.2Sobolev spaces
      3.3Spaces of holomorphic functions
      3.3.1Hardy spaces
      3.3.2Bergman spaces
      4Applications
      4.1Sturm–Liouville theory
      4.2Partial differential equations
      4.3Ergodic theory
      4.4Fourier analysis
      4.5Quantum mechanics
      4.6Color perception
      5Properties
      5.1Pythagorean identity
      5.2Parallelogram identity and polarization
      5.3Best approximation
      5.4Duality
      5.5Weakly-convergent sequences
      5.6Banach space properties
      6Operators on Hilbert spaces
      6.1Bounded operators
      6.2Unbounded operators
      7Constructions
      7.1Direct sums
      7.2Tensor products
      8Orthonormal bases
      8.1Sequence spaces
      8.2Bessel's inequality and Parseval's formula
      8.3Hilbert dimension
      8.4Separable spaces
      9Orthogonal complements and projections
      10Spectral theory
      11See also
      12Remarks
      13Notes
      14References
      15External links


      IP属地:安徽3楼2017-11-01 16:52
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        泛函分析?


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