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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.
A rigged Hilbert space is a pair (H, Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map i is continuous. Identifying H with its dual space H * , the adjoint to i is the map i ∗ : H = H ∗ → Φ ∗ . {\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
The physical significance of the projective Hilbert space is that in quantum theory, the wave functions and represent the same physical state, for any .The Born rule demands that if the system is physical and measurable, its wave function has unit norm, | =, in which case it is called a normalized wave function.
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.
Every separable metric space is homeomorphic to a subset of the Hilbert cube. This is established in the proof of the Urysohn metrization theorem. Every separable metric space is isometric to a subset of the (non-separable) Banach space l ∞ of all bounded real sequences with the supremum norm; this is known as the
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.
This is a Hilbert space under the inner product | = | Given the local nature of our definition, many definitions applicable to single Hilbert spaces apply to measurable families of Hilbert spaces as well. Remark. This definition is apparently more restrictive than the one given by von Neumann and discussed in Dixmier's classic treatise on von ...
Analogous Hamiltonians may be formulated to describe spinless fermions (the Fermi-Hubbard model) or mixtures of different atom species (Bose–Fermi mixtures, for example). In the case of a mixture, the Hilbert space is simply the tensor product of the Hilbert spaces of the individual species. Typically additional terms are included to model ...