Search results
Results from the WOW.Com Content Network
The Hilbertian tensor product of H 1 and H 2, sometimes denoted by H 1 ^ H 2, is the Hilbert space obtained by completing H 1 ⊗ H 2 for the metric associated to this inner product. [87] An example is provided by the Hilbert space L 2 ([0, 1]).
As we have the additional property of the inner product, this is specifically a Hilbert space, because the space is complete under the metric induced by the inner product. This inner product space is conventionally denoted by ( L 2 , ⋅ , ⋅ 2 ) {\displaystyle \left(L_{2},\langle \cdot ,\cdot \rangle _{2}\right)} and many times abbreviated as ...
Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L 2 which is a Hilbert space, or to L 1 and L ∞. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple ...
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.
As any Hilbert space, every space is linearly isometric to a suitable (), where the cardinality of the set is the cardinality of an arbitrary basis for this particular . If we use complex-valued functions, the space L ∞ {\displaystyle L^{\infty }} is a commutative C*-algebra with pointwise multiplication and conjugation.
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 simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions (,).
The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let {} be an orthonormal system in R (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. – see orthogonal polynomials ...