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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
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 ]
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
[4] In his paper of 1947 Segal showed that it is sufficient, for any physical system that can be described by an algebra of operators on a Hilbert space, to consider the irreducible representations of a -algebra. In quantum theory this means that the -algebra is generated by the observables.
As such, quantum states form a ray in projective Hilbert space, not a vector. Many textbooks fail to make this distinction, which could be partly a result of the fact that the Schrödinger equation itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ray is very difficult to avoid. [5]
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.
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 single point denoted in this space is represented by the set of functions () where and represents an index set. In quantum field theory , it is expected that the Hilbert space is also the L 2 {\displaystyle L^{2}} space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined.