TITLE: manifold structure on on a finite dimensional real vector space
QUESTION [16 upvotes]: I am reading Warner's Differentiable Manifolds I do not get one example which is 

Let $V$ be a finite dimensional real vector space. Then $V$ has a natural manifold structure. If $\{e_i\}$ is a basis then the elements of the dual basis $\{r_i\}$ are the coordinate functions of a global coordinate system on $V$.

I don't understand how "the elements of the dual basis $\{r_i\}$ are the coordinate functions of a global coordinate system on $V$." Could any one explain me about that? Then how such a global coordinate system uniquely determines a differentiable structure on $V$? And why this structure is indipendent of choice of basis?
First of all for a manifold structure I need each point must have an open neighborhood $U$ homeomorphic to some open subset of $\mathbb{R}^n$. Here am I getting such notions?

REPLY [11 votes]: The space $\mathbb{R}^n$ has coordinate functions $x_j:\mathbb{R}^n\to\mathbb{R}$, projection onto the $j^{th}$ axis.  If $(\phi,U)$ is a coordinate system on a manifold $M$, then we get coordinate functions on $U$ by composing $\phi$ with the $x_j$.
Warner is just saying that by choosing a basis on a real vector space $V$, you induce a bjiective linear map (hence homeomorphism) $A$ between $V$ and $\mathbb{R}^n$, and that homeomorphism is a global coordinate system with coordinate functions $x_j\circ A = r_j$.  The open neighborhood about each point is the entire space $V$.  
To see that the structure is independent of choice of basis (up to diffeomorphism), try the construction with a different basis; can you see a diffeomorphism between the two structures?