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A complete inner-product space.
A Hilbert space is an inner-product space complete in the induced norm. The Banach-space structure is then canonical and is not an independent input.
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This is an authored directed relation from the source endpoint to the target endpoint.
Require completeness for the norm induced by the inner product.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
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This is an authored directed relation from the source endpoint to the target endpoint.
The inner product induces a norm, and Hilbert completeness makes the resulting normed space a Banach space.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
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Fock space is the graded direct sum of tensor powers of a one-particle Hilbert space.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
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Quotienting nonzero vectors by nonzero complex scalar multiplication gives projective Hilbert space.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
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Standard quantum mechanics represents pure states by rays in a complex Hilbert space and observables by operators acting on it.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
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Hilbert spaces supply state vectors, inner products, and completion for quantum theory.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
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A rigged Hilbert space chooses a dense nuclear or locally convex test space and its dual.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
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The algebraic tensor product is completed in its natural Hilbert norm.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
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