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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 ...
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 Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space () = = = (()) (())Here , the complex scalars, consists of the states corresponding to no particles, the states of one particle, () the states of two identical particles etc.
The direct sum is an operation between structures in abstract ... But every Banach space that is not a Hilbert space necessarily possess some uncomplemented closed ...
For example, the space H can be decomposed as the orthogonal direct sum of two T–invariant closed linear subspaces: the kernel of T, and the orthogonal complement (ker T) ⊥ of the kernel (which is equal to the closure of the range of T, for any bounded self-adjoint operator). These basic facts play an important role in the proof of the ...
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 }.}
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator.