Tensor product

In mathematics, the tensor product ${\displaystyle V\otimes W}$ of two vector spaces ${\displaystyle V}$ and ${\displaystyle W}$ (over the same field) is a vector space to which is associated a bilinear map ${\displaystyle V\times W\rightarrow V\otimes W}$ that maps a pair ${\displaystyle (v,w),\ v\in V,w\in W}$ to an element of ${\displaystyle V\otimes W}$ denoted ⁠${\displaystyle v\otimes w}$⁠.

An element of the form ${\displaystyle v\otimes w}$ is called the tensor product of ${\displaystyle v}$ and ${\displaystyle w}$. An element of ${\displaystyle V\otimes W}$ is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span ${\displaystyle V\otimes W}$ in the sense that every element of ${\displaystyle V\otimes W}$ is a sum of elementary tensors. If bases are given for ${\displaystyle V}$ and ${\displaystyle W}$, a basis of ${\displaystyle V\otimes W}$ is formed by all tensor products of a basis element of ${\displaystyle V}$ and a basis element of ${\displaystyle W}$.

The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from ${\displaystyle V\times W}$ into another vector space ${\displaystyle Z}$ factors uniquely through a linear map ${\displaystyle V\otimes W\to Z}$ (see the section below titled 'Universal property'), i.e. the bilinear map is associated to a unique linear map from the tensor product ${\displaystyle V\otimes W}$ to ${\displaystyle Z}$.

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.