In mathematics, the tensor product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W {\displaystyle V\times W\rightarrow V\otimes W} that maps a pair ( v , w ) {\displaystyle (v,w)} , where v ∈ V , w ∈ W {\displaystyle v\in V,w\in W} , to an element of V ⊗ W {\displaystyle V\otimes W} denoted v ⊗ w {\displaystyle v\otimes w} . An element of the form v ⊗ w {\displaystyle v\otimes w} is called the tensor product of v {\displaystyle v} and w {\displaystyle w} . An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense that every element of V ⊗ W {\displaystyle V\otimes W} is a sum of elementary tensors. If bases are given for V {\displaystyle V} and W {\displaystyle W} , a basis of V ⊗ W {\displaystyle V\otimes W} is formed by all tensor products of a basis element of V {\displaystyle V} and a basis element of W {\displaystyle W} . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V × W {\displaystyle V\times W} into another vector space Z {\displaystyle Z} factors uniquely through a linear map V ⊗ W → Z {\displaystyle V\otimes W\to Z} (see § Universal property), i.e. the bilinear map is associated to a unique linear map from the tensor product V ⊗ W {\displaystyle V\otimes W} to Z {\displaystyle Z} . Tensor products are used in many application areas, including physics and engineering. For example, in general relativity, the gravitational field is described through the metric tensor, which is a tensor field with one tensor at each point of the space-time manifold, and each belonging to the tensor product of the cotangent space at the point with itself.
Definitions and constructions 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 tensor product can also be defined through a universal property; see § Universal property, below. As for every universal property, all objects that satisfy the property are isomorphic through a unique isomorphism that is compatible with the universal property. When this definition is used, the other definitions may be viewed as constructions of objects satisfying the universal property and as proofs that there are objects satisfying the universal property, that is that tensor products exist.
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