Finitely generated projective modules are reflexive

Question

How to show $i_P$ is an isomorphism ? We need to show that it is injective and surjective.

Answer 1

Let's show that $i_P$ is injective. If $i_P x=0$ then for all $f\in P^*$ we have $f(x)=0$ so $x=0$. Hence $i_P$ is injective. Let $F=P\oplus Q$ be a finitely generated free module (which exists since $P$ is projective). Then $i_F:F\rightarrow F^{**}$ is an isomorphism (probably this is covered in the previous page of the book you are citing). Since $i_F$ maps $P$ to $P^{**}$, we deduce that $i_P=i_F\mid_P$ is surjective.