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  2. Hilbert space - Wikipedia

    en.wikipedia.org/wiki/Hilbert_space

    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 ]

  3. Bergman kernel - Wikipedia

    en.wikipedia.org/wiki/Bergman_kernel

    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. Stone's theorem on one-parameter unitary groups - Wikipedia

    en.wikipedia.org/wiki/Stone's_theorem_on_one...

    Stone, M. H. (1930), "Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory", Proceedings of the National Academy of Sciences of the United States of America , 16 (2), National Academy of Sciences: 172– 175, Bibcode : 1930PNAS...16..172S , doi : 10.1073/pnas.16.2.172 , ISSN 0027-8424 , JSTOR 85485 , PMC 1075964 ...

  5. Mathematical formulation of quantum mechanics - Wikipedia

    en.wikipedia.org/wiki/Mathematical_formulation...

    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]

  6. Representer theorem - Wikipedia

    en.wikipedia.org/wiki/Representer_theorem

    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.

  7. Direct integral - Wikipedia

    en.wikipedia.org/wiki/Direct_integral

    The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X.Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions

  8. Hilbert's thirteenth problem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_thirteenth_problem

    Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question when posed for the class of continuous functions.

  9. Quantum configuration space - Wikipedia

    en.wikipedia.org/wiki/Quantum_configuration_space

    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.