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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 ...
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.
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 (algebraic) direct sum of ... The fact that there is a unique separable Hilbert space has a ... and be two Hilbert modules over the same C ...
The two other theorems are Hilbert's basis theorem, ... denote the direct sum of ... which is an algebraic formulation of the fact that affine n-space is a variety ...
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 }.}