Are co and contravariant relative or absolute properties? Given is a vector space $V$ and a dual vectorspace $V^*$. Elements of the vector space are typically decomposed as linear combinations of contravariant components times covariant basis vectors,
$$\vec v = v^i \vec e_i. $$
The elements of the dual vectorspace are linearcombinations of covariant components times contravariant basis vectors,
$$\hat w = w_i \hat e^i.$$
But couldn't i have switched all the names ? Wasn't the assignment of the prependices co/contra solely based on my decision to designate the first space as the "normal" vectorspace and the second as its dual?
Assuming i was given only $V^*$  could i determine that this is a dual space and that the elements should  be decomposed in a contravariant basis instead of a covariant basis ?
 A: It is not easy to understand what you want to know.
The concepts of "covariant" and "contravariant" are related to the behavior under a change of basis of a vector space $W$. See my answer to Why is tensor from a vector space covariant, not contravariant?
These concepts only make sense for finite-dimensional vector spaces. If we are given a basis $\{e_i\}$ of $W$, the dual vectors $e^*_i$ only form a basis of $W^*$ if $\dim W < \infty$.
If we look at both $V$ and $V^*$, we consider bases for $V$ and their dual bases for $V^*$. These are the fundaments for denoting basis vectors resp. components as covariant and contravariant.
Of course we may change perspective and consider bases of the abstract vector space $W = V^*$. Then we get decompositions of vectors as linear combinations of contravariant components times covariant basis vectors. But this means that the starting point is no longer $V$ with its bases but $W = V^*$ with its bases. A basis transformation on $W$ is dual to a basis transformation on $V$, and this flips "covariant" and "contravariant".
