Are tensors multilinear maps or are they elements of the domain of these maps? In other words, are tensors operators or the objects acted on by the operators?

Answer 1

A tensor is a multilinear map on vectors and covectors. Specifically, if V is a finite dimensional real vector space, an (n,m)-tensor on V is defined as a multilinear map T : (V*)^n x V^m -> R, where (V*)^n = V* x V* x ... x V* (n times) and V^m = V x V x ... x V (m times). In particular, (1,1)-tensors on V, i.e., bilinear maps T : V* x V -> R, may be viewed as linear operators on V.

An alternative, but equivalent definition of an (n,m)-tensor on V is an element of the tensor product space V^⊗n ⊗ (V*)^⊗m , where V^⊗n is the n-fold tensor product V ⊗ V ⊗ ••• ⊗ V (n times) and (V*)^⊗m is the m-fold tensor product V* ⊗ V* ⊗ ••• ⊗ V* (m times). Note, the tensor products are taken over the reals.