nLab tensor product of vector spaces

Idea

The tensor product of two vector spaces is a new vector space with the property that bilinear maps out of the Cartesian product of the two spaces are equivalently linear maps out of the tensor product.

The tensor product of vector spaces is just the special case of the tensor product of modules over some ring $R$ for the case that this ring happens to be a field.

The tensor product of vector spaces makes the category Vect of all vector spaces into a monoidal category, in fact a distributive monoidal category.

Definition

Definition

Given two vector spaces over some field $k$, $V_1, V_2 \in Vect_k$, their tensor product of vector spaces is the vector space denoted

$$V_1 \otimes_k V_2 \in Vect$$

whose elements are equivalence classes of formal linear combinations of tuples $(v_1,v_2)$ with $v_i \in V_i$, for the equivalence relation given by

$$(k v_1 , v_2) \;\sim\; k( v_1 , v_2) \;\sim\; (v_1, k v_2)$$

$$(v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2)$$

$$(v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2)$$

More abstractly this means that the tensor product of vector spaces is the vector space characterized by the fact that

1. it receives a bilinear map

   $$V_1 \times V_2 \longrightarrow V_1 \otimes V_2$$

   (out of the Cartesian product of the underlying sets)

2. any other bilinear map of the form

   $$V_1 \times V_2 \longrightarrow V_3$$

   factors through the above bilinear map via a unique linear map

   $$\array{ V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3 \\ \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}} \\ V_1 \otimes_k V_2 }$$

References

• Tensor Products, §III.89 in: The Princeton Companion to Mathematics (2009) [doi:10.1515/9781400830398.301a]

Related concepts

• tensor product of modules

• tensor product of chain complexes

• tensor product of vector bundles

• graded vector space

• inductive tensor product