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.