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A Hilbert space is a vector space equipped with an inner product operation, which allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.
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One key observation about this picture is that L 2,h (D) may be identified with the space of holomorphic (n,0)-forms on D, via multiplication by . Since the L 2 {\displaystyle L^{2}} inner product on this space is manifestly invariant under biholomorphisms of D, the Bergman kernel and the associated Bergman metric are therefore automatically ...
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:, is continuous. [ 4 ] [ 5 ] Identifying H with its dual space H * , the adjoint to i is the map i ∗ : H = H ∗ → Φ ∗ . {\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
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 ...
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 ...
The closed tangent cone to the cube at the zero vector is the whole space. Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the ...
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 ...