Covariance and contravariance Given a vector space $V$, a vector $v \in V$ can be written in components with respect to different bases, say $X$ and $Y$. Now when i make a transformation from $X$ to $Y$, the components of the vector are transforming contravariantly.
Now the dual space$V^*$ of $V$ is also a vector space, but the components of a vector there transform differently in a change of dual basis, i.e. covariantly.
My question is, if we see the dual space $V^*$ as a vector space $W$, having no relation with the vector space $V$, will we then say that the components of a vector $w \in W$ will transform contravariantly?

The condition "$V$ left or right vector space over a field $F$" is purely algebraic. It tells us nothing on the type of transformations of tensors on it. That is actually due to further definitions, i.e. those of vectors/vector fields and covectors/covector fields on $V$. In other words, being a vector space doesn't imply that vectors transform "covariantly" or "contravariantly". Vectors just transform according to a particular law which involves a matrix $J$, if you change basis. Correspondingly, covectors change according to a law which involves $J^{-1}$, and this is a consequence of the definition of dual basis. Because of this duality, we give these two types of transformations two different names. You can see how one comes after each other, they are not intrinsic properties of the structure of vector space.
