PlanetPhysics/Hilbert Space 3

Basic concepts

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An inner-product space with complex scalars,  , is a vector space   with complex scalars, together with a complex-valued function  , called the inner product, defined on  , which has the following properties:

  • (1) For all  .
  • (2) If   then  .
  • (3) For all   and   in  ,  .
  • (4) For all   and   in  ,  .
  • (5) For all   in V, and all scalars  , one has that  .(The inner product is linear in the first variable, and conjugate linear in the second.)

A Banach space   is a normed vector space such that   is complete under the metric induced by the norm  .

Hilbert space

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A Hilbert space is an inner product space which is complete as a metric space, that is for every sequence   of vectors in  , if   as   and   both tend to infinity, there is in  , a vector   such that   as  . (In quantum physics, all Hilbert spaces are tacitly assumed to be infinite dimensional)

Remarks

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Sequences with the property that   are called Cauchy sequences . Usually one works with Hilbert spaces because one needs to have available such limits of Cauchy sequences. Finite dimensional inner product spaces are automatically Hilbert spaces. However, it is the infinite dimensional Hilbert spaces that are important for the proper foundation of quantum mechanics.

A Hilbert space is also a Banach space in the norm induced by the inner product, because both the norm and the inner product induce the same metric.