# adjoint map and dual map of complex inner product space

I know (a). but I can't solve (b) and (c).

Can you help me please?

## 1 Answer

(b) For a given $w$, the map $v\mapsto\langle Tv,w\rangle$ is a linear functional, so you can apply (a) to obtain $T^*w$.

(c) For a $g\in W^*$, i.e. $g:W\to\Bbb C$ linear, then $T^tg=v\mapsto g(Tv)$ by definition. Applying (a) again, we get that $g=\langle -,u\rangle$, namely with $u=\phi_W(g)$. So, $T^t\phi_W^{-1}u=v\mapsto g(Tv)=v\mapsto \langle Tv,u\rangle$, but by (b) that equals to $\langle v,T^*u\rangle$, arriving to the map $\,\phi_V^{-1}T^*u$.