Are there cases in which we should consider tensors as equivalence classes?

Usually in Physics texts in which tensors ared used, one defines them as multilinear maps. So if $$V$$ is a vector space over the field $$F$$, a tensor is a multilinear mapping:

$$T:V\times\cdots\times V\times V^\ast\times\cdots\times V^\ast\to F.$$

In texts about multilinear algebra, however, a tensor is defined differently. They consider a collection $$V_1,\dots,V_k$$ of vector spaces over the same field, consider the free vector space $$\mathcal{M}=F(V_1\times\cdots\times V_k)$$, consider the subspace $$\mathcal{M}_0$$ genereated by vectors of the form

$$(v_1,\dots,v_i+v_i',\dots,v_k)-(v_1,\dots,v_i,\dots,v_k)-(v_1,\dots,v_i',\dots,v_k),$$

$$(v_1,\dots,kv_i,\dots,v_k)-k(v_1,\dots,v_i,\dots,v_k),$$

and then define the tensor product $$V_1\otimes\cdots\otimes V_k = \mathcal{M}/\mathcal{M}_0$$ and define tensors as elements of such space, which are equivalence classes of functions with finite support in $$V_1\times\cdots\times V_k$$.

Now, is there some cases in Physics where it's better to think as tensors as such equivalence classes rather than multilinear mappings? If so, how then we get some physical intuition behind those objects?

• I think you just need to understand that the two constructions give isomorphic spaces.
– MBN
Commented Oct 30, 2013 at 17:50