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Usually in texts about Physics that uses tensors 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?

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    $\begingroup$ I think you just need to understand that the two constructions give isomorphic spaces. $\endgroup$ – MBN Oct 30 '13 at 17:50
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The second definition you provided is more useful to "Algebraists" rather than to Physicists, but in recent times there are few communications between Number Theory and Statistical Physics, for say in Bost-Connes system or Ruelle's Lee-Yang system, people there use finite or infinite lattices to define "Partition Functions" of the Quantum Statistical Systems, actually there you need the second definition of "Tensor Product" as you might have to "Extend" the theory defined over some Module to some other Module, actually the second definition of Tensor Product is inspired from Galois Theory mainly, a useful reference would be "Bost, J.-B.; Connes, Alain (1995), "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Mathematica. New Series 1 (3): 411–457, doi:10.1007/BF01589495 , ISSN 1022-1824 , MR 1366621 " for an application of the above ideas.

Also, Penrose-Rindler's book "Spinors and Space-Time" gives a very Geometric as well as Physical intuition behind the first definition of Tensor and their products.

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