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A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel K(ζ, z) that verifies a reproducing property analogous to this one. The Hardy space H 2 ( D ) also admits a reproducing kernel, known as the Szegő kernel . [ 37 ]
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.
Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinite-dimensional Hilbert space which is closed and bounded but not weakly compact since it doesn't contain 0). However, bounded and weakly closed sets are weakly compact so as a consequence every ...
A feature map is a map :, where is a Hilbert space which we will call the feature space. The first sections presented the connection between bounded/continuous evaluation functions, positive definite functions, and integral operators and in this section we provide another representation of the RKHS in terms of feature maps.
Isometries preserve Cauchy sequences; hence the completeness property of Hilbert spaces is preserved [3] The following, seemingly weaker, definition is also equivalent: Definition 3. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: the range of U is dense in H, and
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its Hermitian adjoint N*, that is: NN* = N*N. [1] Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood.
The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal ...
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). [4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with ...