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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 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.
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.
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 ...
This representation is called outer tensor product of the representations and . The existence and uniqueness is a consequence of the properties of the tensor product. Example. We reexamine the example provided for the direct sum:
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above.
For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides ...
the vertex set of G × H is the Cartesian product V(G) × V(H); and; vertices (g,h) and (g',h' ) are adjacent in G × H if and only if. g is adjacent to g' in G, and; h is adjacent to h' in H. The tensor product is also called the direct product, Kronecker product, categorical product, cardinal product, relational product, weak direct product ...