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Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space.
For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group on . The infinitesimal generator of this group is the system Hamiltonian .
Moore was interested in generalization of integral equations and showed that to each such there is a Hilbert space of functions such that, for each , = (, (,)). This property is called the reproducing property of the kernel and turns out to have importance in the solution of boundary-value problems for elliptic partial differential equations.
For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.
Let denote a random variable with domain and distribution .Given a symmetric, positive-definite kernel: the Moore–Aronszajn theorem asserts the existence of a unique RKHS on (a Hilbert space of functions : equipped with an inner product , and a norm ‖ ‖) for which is a reproducing kernel, i.e., in which the element (,) satisfies the reproducing property
The local index formula [2] expresses the pairing of the K-group of the manifold with this K-cycle in two ways: the 'analytic/global' side involves the usual trace on the Hilbert space and commutators of functions with the phase operator (which corresponds to the 'index' part of the index theorem), while the 'geometric/local' side involves the ...
[4] There have been three main ways to formulate spectral theory, each of which find use in different domains. After Hilbert's initial formulation, the later development of abstract Hilbert spaces and the spectral theory of single normal operators on them were well suited to the requirements of physics, exemplified by the work of von Neumann. [5]
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra. Thus R 0 {\displaystyle R_{0}} is an Artinian ring , which is a k -vector space of dimension P (1) , and Jordan–Hölder theorem may be used for proving that P (1) is the degree of the ...