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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.
In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.
In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product.
K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit. Ab, the category of abelian groups, with the group of integers Z serving as the unit. For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit.
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m , then their outer product is an n × m matrix.
Kronecker product, the tensor product of matrices (or vectors), which satisfies all the properties for vector spaces and allows a concrete representation; Tensor product of Hilbert spaces, endowed with a special inner product as to remain a Hilbert space Other topological tensor products; Tensor product of graphs, an operation on graphs, whose ...
In differential geometry, the tensor product of vector bundles E, F (over same space ) is a vector bundle, denoted by E ⊗ F, whose fiber over a point is the tensor product of vector spaces E x ⊗ F x. [1] Example: If O is a trivial line bundle, then E ⊗ O = E for any E.
For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense, [29] and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces. [30] A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more general sense.