I am studying relativity, and as you know the theory extensively uses the notion of covariant and contravariant component of vectors.
My question is the following. Let say $\vec{x}$ is a vector which belongs to vector space V with basis vectors ${e}_{i}$. We know that $\vec{x}$ can be written as: ${x}^{i} {e}_{i}$. The same vector $\vec{x}$ can be written in terms of the basis vectors ${e}^{i}$ of the dual space V* as: ${x}_{i} {e}^{i}$, why is this true? Isn't ${e}^{i}$ a basis set for the dual space V* and not the vector space V? How can the same vector belong to both the vector space V and its dual V*? How can $\vec{x}$ which belongs to V be written in terms of the basis vectors of another vector space V*?
The Wikipedia article on Covariant transformation says that:
...so the dual space has the same dimension as the linear space itself. It is "almost the same space", except that the elements of the dual space (called dual vectors) transform covariantly and the elements of the tangent vector space transform contravariantly.
What does "almost the same" mean?