Tensor operation on a vector space From the various definitions provided in the article https://en.wikipedia.org/wiki/Tensor, the tensor seems always to be defined, even in the more abstract forms, as a multilinear map, from a product of vector and dual spaces to the underlying field.
However, in applied mathematics, one often come across a tensor when used in the form that maps elements of vector and dual spaces to elements of vectors and dual spaces, like this, for example:
$\theta^j=\mu_l^j dx^l$
No operation is defined in the article between a tensor and a dual vector, which gives a dual vector. But since other operations are defined on tensors, should the previous notation actually be read in two steps:


*

*a tensor product: $\mu_l^j dx^k$

*followed by a contraction of the indices $k$ and $l$


Yes. This is what is called the Einstein summation convention. 
(I happen to think it is ultimately less confusing to define tensors as elements of tensor products of a vector space and its dual, but whatever floats your boat, I suppose.) 
