I have no idea what dual space and dual map are, so have much trouble understanding cotangent bundle when reading Lee's book.
First of all, can anyone give me a introduction what the dual map and dual space are? I googled it, but still not sure I understand it.
Then, there is a remark saying:
Even though for each $p\in M$, the map $T_pf:T_pM\rightarrow > T_{f(p)}N$ has a dual map $(T_pf)^*:(T_{f(p)}N)^*\rightarrow (T_pM)^*$, these maps do not generally combine to give a map from $T^*M$ to $T^*N$.
I do not understand it, maybe it's because I do not understand dual map.
Then Lee proved the following thing:
$\left((T_p\text{x})^{-1}\right)^*(\theta_p)\cdot(v)=\theta\left((T_P\text{x})^{-1}\cdot v\right)$
where $\text{x}$ is a chart map, $\theta_p\in T_p^*M$ and $v\in T_PM$.
I don't understand his argument. Particularly, he said:
$ \left((T_p\text{x})^{-1}\right)^*(\theta_p)\cdot(v)=\theta_p\left((T_P\text{x})^{-1}\cdot v\right) $
Why?
What is $\left((T_p\text{x})^{-1}\right)^*$?