Relation between Vector space $V$ and its dual $V^{*}$ I asked the same question in Math.SE, but I was suggested to ask it here as well.
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 it's 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?
Again, along the same question, you see that in the Wikipedia article on Curvilinear coordinates, 
both bases (vector basis $e_{i}$ and covector basis $e^{i}$) are plotted in the same graph, although we know that they are bases for different vector spaces. 
 A: If $V$ is a (finite dimensional) vector space with no additional structure, then $V^*$ is a different vector space of the same dimension, hence isomorphic to $V$ --- but it would be a great mistake to think of $V$ and $V^*$ as "the same'' or even ``almost the same'', because there is no preferred isomorphism between them.  In other words, you can pick a basis for $V$, pick a basis for $V^*$, map the elements of the one basis to the other, and get an isomorphism, but there a gazillion different ways to do this, and no reason to prefer one to another.  So there is no single natural way to identify elements of $V$ with elements of $V^*$.
On the other hand, if $V$ is not just a vector space, but a vector space equipped with a metric, then there is one very natural way of identifying $V$ with $V^*$ ---- first pick an orthogonal basis for $V$ (note that the very notion of ``orthogonal'' makes no sense without the extra structure), then map the elements of that basis to the corresponding elements of the  dual basis, and then check that you'll get exactly the same map no matter which orthogonal basis you started with.  So in this case, $V$ and $V^*$ are "almost the same" in the sense that there is one preferred way to identify an element of $V$ with an element of $V^*$.
Of course in the usual applications to relativity, your vector space comes equipped with a metric, so the second paragraph above applies instead of the first.
A: 
The same vector x⃗  can be written in terms of the basis vectors ei of
  the dual space V* as: xiei, why is this true?

It's not true.
The elements of the dual space are not vectors as we ordinarily conceive of them geometrically, e.g., directed line segments.
Rather, they are (geometrically) a set of oriented surfaces with a density proportional to their magnitude.
For example, consider a vector in the $x$ direction of length $L$.  The associated element of the dual space, a one-form, is the set of surfaces in the $yz$ plane with density $L$.
The contraction of a vector and its dual gives the real number $L^2$ and, geometrically, is the number of surfaces of the one-form pierced by the vector.
For unit basis one-forms, the density is 1.
Now, picture graph paper with the coordinate axes drawn:

The vertical surfaces represent the one-form dual to the $x$ basis vector while the horizontal surfaces represent the one-form dual to the $y$ basis vector.
Draw a vector from the origin to the point, e.g., (3,4).  The vector is then given by
$$3\hat{\mathbf x} + 4\hat{\mathbf y}$$
Note that the $x$ and $y$ components are precisely the numbers given by the contraction of the vector with the respective basis one-forms.
In other words, the vector pierces 3 vertical surfaces and 4 horizontal surfaces.
For cartesian coordinates in a flat space, a vector and its dual one-form have the same components.  Thus, one might be led to falsely conclude that they represent the same object - they don't.  Generally, the vector and its dual one-form do not have the same components.
Recalling matrix representation, a one-form is represented by a row vector while a vector is represented by a column vector.  The row vectors 'live' in a different space than the column vectors.
