Hilbert space dimension | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Hilbert_space


00:02:50 1 Definition and illustration
00:03:00 1.1 Motivating example: Euclidean space
00:08:50 1.2 Definition
00:10:07 1.3 Second example: sequence spaces
00:17:05 2 History
00:18:50 3 Examples
00:22:10 3.1 Lebesgue spaces
00:26:10 3.2 Sobolev spaces
00:26:20 3.3 Spaces of holomorphic functions
00:29:23 3.3.1 Hardy spaces
00:33:07 3.3.2 Bergman spaces
00:33:17 4 Applications
00:34:50 4.1 Sturm–Liouville theory
00:38:37 4.2 Partial differential equations
00:39:17 4.3 Ergodic theory
00:41:57 4.4 Fourier analysis
00:45:27 4.5 Quantum mechanics
00:50:22 4.6 Color perception
00:58:02 5 Properties
01:00:39 5.1 Pythagorean identity
01:01:32 5.2 Parallelogram identity and polarization
01:01:41 5.3 Best approximation
01:05:02 5.4 Duality
01:07:52 5.5 Weakly-convergent sequences
01:10:30 5.6 Banach space properties
01:13:18 6 Operators on Hilbert spaces
01:15:10 6.1 Bounded operators
01:17:08 6.2 Unbounded operators
01:18:38 7 Constructions
01:18:48 7.1 Direct sums
01:20:50 7.2 Tensor products
01:24:25 8 Orthonormal bases
01:26:32 8.1 Sequence spaces
01:26:41 8.2 Bessel's inequality and Parseval's formula
01:31:31 8.3 Hilbert dimension
01:34:44 8.4 Separable spaces
01:35:53 9 Orthogonal complements and projections
01:36:41 10 Spectral theory
01:37:13 11 In popular culture
01:39:56 12 See also
01:40:44 13 Remarks
01:44:40 14 Notes
01:46:17 15 References
01:48:13 16 External links
01:49:52 PV. Moreover, any self-adjoint linear operator E such that E2
01:50:29 0. This is equivalent to U and V being orthogonal as subspaces of H. The sum of the two projections PU and PV is a projection only if U and V are orthogonal to each other, and in that case PU + PV
01:54:58 Spectral theory
02:04:10 In popular culture
02:04:55 See also
02:05:20 Remarks



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SUMMARY
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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 specif ...