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The direct sum is a submodule of the direct product of the modules M i (Bourbaki 1989, §II.1.7). The direct product is the set of all functions α from I to the disjoint union of the modules M i with α(i)∈M i, but not necessarily vanishing for all but finitely many i. If the index set I is finite, then the direct sum and the direct product ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written as + the phrase "direct sum" is used, while if the group operation is written the phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the ...
The tensor product, or simply , of two vector spaces and is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map g : V × W → X {\displaystyle g:V\times W\to X} from the Cartesian product V × W {\displaystyle V\times W} is called bilinear if g {\displaystyle g ...
The direct sum and direct product are not isomorphic for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory : the direct sum is the coproduct , while the direct product is the product.
Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation ( V ⊕ W ) i = V i ⊕ W i . If I is a semigroup , then the tensor product of two I -graded vector spaces V and W is another I -graded vector space, V ⊗ W {\displaystyle V\otimes W} , with gradation
The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring .
Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the n-particle states are vectors in a symmetrized tensor product of n single-particle Hilbert spaces H.