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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 ]
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 ...
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 the Hilbert–Schmidt norm). [4] The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional.
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1] It is a result of studies of linear algebra and the solutions of systems of linear equations and their ...
The sesquilinear form B : H × H → is separately uniformly continuous in each of its two arguments and hence can be extended to a separately continuous sesquilinear form on the completion of H; if H is Hausdorff then this completion is a Hilbert space. [1] A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.
The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the strong operator topology are precisely those that are continuous in the WOT (actually, the WOT is the weakest operator topology that leaves continuous all strongly continuous linear functionals on the set () of bounded operators on the Hilbert ...
Let be an arbitrary set and a Hilbert space of real-valued functions on , equipped with pointwise addition and pointwise scalar multiplication.The evaluation functional over the Hilbert space of functions is a linear functional that evaluates each function at a point ,
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit ...