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. $\{e_i\}$ is a basis then the elements of the dual basis $\{r_i\}$ are the coordinate functions of a global coordinate system on $V$, here 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 a 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 nbd $U$ homeomemorphic to some open subset of $\mathbb{R}^n$, Here am I getting such notions?
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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 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? |
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