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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 ]
This function is a test function on and is an element of (). The support of this function is the closed unit disk in . It is non-zero on the open unit disk and it is equal to 0 everywhere outside of it.
where H(D) is the space of holomorphic functions in D. Then L 2, h ( D ) is a Hilbert space: it is a closed linear subspace of L 2 ( D ), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D
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.
It is clear from the definition of the inner product on the GNS Hilbert space that the state can be recovered as a vector state on . This proves the theorem. This proves the theorem. The method used to produce a ∗ {\displaystyle *} -representation from a state of A {\displaystyle A} in the proof of the above theorem is called the GNS ...
The first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by
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 ...
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