Hilbert Space
A Hilbert Space is a topological vector space over the complex numbers.
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- Example(s):
- See: Metric Space, Banach Space, Normed Space, Normed Linear Space.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Hilbert_space Retrieved:2015-6-7.
- 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.
- 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.
2012
- (Venkatachalaiyer, 2012) ⇒ Alampallam Balakrishnan Venkatachalaiyer. (2012). “Introduction to Optimization Theory in a Hilbert Space." Vol. 42. Springer Science & Business Media,
- QUOTE: … A normed linear space is complete if every Cauchy sequence converges (in norm) to an element in the space. Def. A Hilbert Space is a complete inner product space. We note that every normed linear space can be completed. …
2004
- (Lanckriet et al., 2004a) ⇒ Gert R. G. Lanckriet, Nello Cristianini, Peter Bartlett, Laurent El Ghaoui, and Michael I. Jordan. (2004). “Learning the Kernel Matrix with Semidefinite Programming.” In: The Journal of Machine Learning Research, 5.
- QUOTE: Kernel-based learning algorithms (see, for example, Cristianini and Shawe-Taylor, 2000; Scholkopf and Smola, 2002; Shawe-Taylor and Cristianini, 2004) work by embedding the data into a Hilbert space, and searching for linear relations in such a space. The embedding is performed implicitly, by specifying the inner product between each pair of points rather than by giving their coordinates explicitly.
1997
- (Luenberger, 1997) ⇒ David G. Luenberger. (1997). “Optimization by Vector Space Methods." Wiley Professional. ISBN:047118117X
1970
- (Pollingher & Zaks, 1970) ⇒ Adolf Pollingher, and Abraham Zaks. (1970). “On Baer and Quasi-Baer Rings." Duke Mathematical Journal 37, no. 1
- QUOTE: … generated by an idempotent. The motivation comes from the observation that the theory of rings of operators on a Hilbert space is a particular case of a pure algebraic theory of Baer rings satisfying some axioms. …
1964
- (Nussbaum, 1964) ⇒ A E Nussbaum. (1964). “Reduction Theory for Unbounded Closed Operators in Hilbert Space." Duke Mathematical Journal 31, no. 1
- QUOTE: … matrix o] the operator A. Since a projection P in a Hilbert space is a bounded self-adjoint idempotent operator, it follows that P is a projection in 3C X 3C if and only if the elements of the matrix (P.) satisfy the relations (i) P*=P. and …
1955
- (Bram, 1955) ⇒ Joseph Bram. (1955). “Subnormal Operators." Duke mathematical journal 22, no. 1
- QUOTE: … Throughout this paper, a Hilbert space is a vector space over the complex numbers, an operator is a bounded linear transformation, and a subspace is a closed linear manifold. ...