Problems about dual map, cotangent bundle.

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 $\left((T_p\text{x})^{-1}\right)^*$?

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First, do you understand the concept of the dual of a (finite dimensional) vector space? If not, try googling some lecture notes on it –  gabbering Oct 24 '12 at 11:53
I think I know, it is a linear functional map a vector space to $\mathbb{R}$?...Ok..I will try to find some notes. –  hxhxhx88 Oct 24 '12 at 12:25