I have two questions to ask about:
For example I have a tensor $T(r,s)$ by which it means that $T$ operates on $r$ vectors and $s$ dual vectors. Take for example $T(3,0)$, so in this notation it only operates on three vectors. Then I heard that by raising or lowering indices one can manipulate the number of vectors and dual vectors a tensor can act on without affecting the value. In the above example, it means, e.g. $T(3,0) = T(2,1) = T(1,2) = T(0,3)$ (note that those four notations are usually written with differing notation for $T$ because they are now different object if a lowering or raising index has been done, but I don't bother to make them have different notation just for my convenience). Are those equalities true?
I just want to check if my understanding about the definition of tensor is correct. Tensor is any multilinear operators/functions which return a scalar. An example of a tensor is the inner product in real vector space. However, inner product in a complex vector space is not a tensor because it's conjugate linear (i.e. not linear) in one of the arguments. Is it right?
EDIT: I reworded my questions to make it clear that I have two questions now. The first one has been unofficially answered as it is written as a comment.