Tensor (intrinsic Definition) - Universal Property

Universal Property

The space can be characterized by a universal property in terms of multilinear mappings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for free modules, and the "universal" approach carries over more easily to more general situations.

A scalar-valued function on a Cartesian product (or direct sum) of vector spaces

is multilinear if it is linear in each argument. The space of all multlinear mappings from the product V₁×V₂×...×V_N into W is denoted LN(V₁,V₂,...,V_N; W). When N = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V;W).

The universal characterization of the tensor product implies that, for each multilinear function

there exists a unique linear function

such that

for all v_i ∈ V and α_i ∈ V∗.

Using the universal property, it follows that the space of (m,n)-tensors admits a natural isomorphism

In the formula above, the roles of V and V* are reversed. In particular, one has

and

and