Search results
Results from the WOW.Com Content Network
The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutually orthogonal subspaces. If infinitely many Hilbert spaces H i {\displaystyle H_{i}} for i ∈ I {\displaystyle i\in I} are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be ...
The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system.
In mathematics and functional analysis, a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of ...
Unlike the case of the circle, H 2 − (∂Ω) and H 2 + (∂Ω) are not orthogonal spaces. By the Hartogs−Rosenthal theorem, their sum is dense in L 2 (∂Ω). As shown below, these are the ±i eigenspaces of the Hilbert transform on ∂Ω, so their sum is in fact direct and the whole of L 2 (∂Ω).
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 ...
Because is closed and is a Hilbert space, [note 4] can be written as the direct sum = [note 5] (a proof of this is given in the article on the Hilbert projection theorem). Because K ≠ H , {\displaystyle K\neq H,} there exists some non-zero p ∈ K ⊥ . {\displaystyle p\in K^{\bot }.}
An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both.
Notice how this generalises the special case of the one-dimensional Hilbert space, where U(C) is just the circle group.) Given these definitions, we can state the second part of the Peter–Weyl theorem (Knapp 1986, Theorem 1.12): Peter–Weyl Theorem (Part II). Let ρ be a unitary representation of a compact group G on a complex Hilbert space H.