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×...×VN into W is denoted LN(V1,V2,...,VN; 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 vi ∈ V and αi ∈ 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
Read more about this topic: Tensor (intrinsic Definition)
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