direct sum of Hilbert spaces

Let {Hi}iI be a family of Hilbert spacesMathworldPlanetmath indexed by a set I. The direct sumMathworldPlanetmathPlanetmath of this family of Hilbert spaces, denoted as


consists of all elements v of the Cartesian productMathworldPlanetmath ( of {Hi}iI such that vi2<. Of course, for the previous sum to be finite only at most a countableMathworldPlanetmath number of vi can be non-zero.

Vector addition and scalar multiplication are defined termwise: If u,viIHi, then (u+v)i=ui+vi and (sv)i=svi.

The inner productMathworldPlanetmath of two vectors is defined as


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Title direct sum of Hilbert spaces
Canonical name DirectSumOfHilbertSpaces
Date of creation 2013-03-22 14:43:55
Last modified on 2013-03-22 14:43:55
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Definition
Classification msc 46C05
Related topic CategoryOfHilbertSpaces