Let $V=P_1(\mathbb{R})$ and $p \in V$. Define $f_1(p)=\int_0^1 p(t) dt$ and $f_2(p)=\int_0^2 p(t) dt$. Show that $\{f_1,f_2\}$ is a basis for $V^*$, and find the basis for the dual. Next find the basis for $V^{**}$ that is dual to $\{f_1,f_2\}$.
I have found the basis for the dual, which is $\{2-2x,x-1/2\}$. Next I have to show that the $\{f_1,f_2\}$ is linearly independent and spans $V^*$. I'm getting confused in trying to argue that this set is in fact a basis. So far I have: to show that it spans, I have: $\forall f \in V^* \exists\alpha, \beta \in F$ s.t. $f=\alpha f_1(p)+\beta f_2(p)$.
Then for the set to be linearly independent, suppose that $\exists \alpha \neq 0, \beta \neq 0$ s.t. $\alpha f_1(p)+\beta f_2(p)$, then proceed to show a contradiction, so then we have that $\{f_1,f_2\}$ is a basis of $V^*$.
I'm unsure how to go about finding a basis for $V^{**}$, and would very much appreciate any help in finding the basis. Thanks in advance.