My question is as it says in the title really. I've been reading Nakahara's book on geometry and topology in physics and I'm slightly stuck on a part concerning adjoint mappings between vector spaces. It is as follows:
Let $W=W\left(n, \mathbb{R}\right) $ be a vector space with a basis $\lbrace\mathbf{f}_{\alpha}\rbrace$ and a vector space isomorphism $G:W\rightarrow W^{\ast}$. Given a map $f:V\rightarrow W$, we may define the adjoint of $f$, denoted by $\tilde{f}$, by $$G\left(\mathbf{w}, f\mathbf{v}\right) =g\left(\mathbf{v}, \tilde{f}\mathbf{w}\right) $$ where $\mathbf{v} \in V$ and $\mathbf{w} \in W$, and $g(\cdot,\cdot) $ is the inner product between the two vectors $\mathbf{v} $ and $\tilde{f}\mathbf{w}$.
He then goes on to say that "it is easy to see from this", that $\widetilde{(\tilde{f})}=f$.
I'm having trouble showing that this is true given the definitions above.