I have just starting learning relativity and I have still a problem with the definition and notation of tensors. So a (r,s)-tensor takes r vectors, s one-forms and gives a scalar. Then I understand that a (1,0)-tensor takes 1 vector (e.g. from V) and gives a scalar which is exactly the definition of a one-form (in V*), which corresponds to the mapping (V->R). But I am still uncomfortable with the symmetrical situation, i.e. that a (0,1)-tensor is a vector. A (0,1)-tensor takes a one-form (e.g. u \in V*) and gives a scalar. But the one-form u given as argument is itself a mapping from V to R, so in a sense my (0,1)-tensor is a mapping ((V -> R) -> R) and instinctively I would "reduce" it to (V->R) or (V->RxR) but I cannot figure out how at the end it gives something which is again in V. I am probably wrong in the vector spaces I consider, am I ? Thanks a lot for your help.