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  2. Hilbert space - Wikipedia

    en.wikipedia.org/wiki/Hilbert_space

    Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.

  3. Weak convergence (Hilbert space) - Wikipedia

    en.wikipedia.org/wiki/Weak_convergence_(Hilbert...

    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 ...

  4. Mathematical formulation of quantum mechanics - Wikipedia

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

    The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

  5. Wave function - Wikipedia

    en.wikipedia.org/wiki/Wave_function

    The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces. [ nb 10 ] Its basis is { e i , i ∈ Z } with e i ( j ) = δ ij , i , j ∈ Z . The most basic example of spanning polynomials is in the space of square integrable functions on the interval [–1, 1] for which the Legendre polynomials is a Hilbert ...

  6. Reproducing kernel Hilbert space - Wikipedia

    en.wikipedia.org/wiki/Reproducing_kernel_Hilbert...

    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.

  7. Quantum configuration space - Wikipedia

    en.wikipedia.org/wiki/Quantum_configuration_space

    In quantum field theory, it is expected that the Hilbert space is also the space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined. However, it is often hard to define a concrete Borel measure on the classical configuration space, since the integral theory on infinite ...

  8. Quantum state - Wikipedia

    en.wikipedia.org/wiki/Quantum_state

    Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.

  9. Quantum state space - Wikipedia

    en.wikipedia.org/wiki/Quantum_state_space

    In quantum mechanics a state space is a separable complex Hilbert space.The dimension of this Hilbert space depends on the system we choose to describe. [1] [2] The different states that could come out of any particular measurement form an orthonormal basis, so any state vector in the state space can be written as a linear combination of these basis vectors.

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