This is an extension of theorem 6.8 of Friedberg Linear Algebra.
Let $V$ be a finite dimensional complex inner product space. I know for every linear functional $f$ on $V$, $\exists$ a unique vector $w \in V$ such that $f(v) = \langle v, w\rangle$. [This is theorem 6.8 in Friedberg]
Now from this, I want to know how to prove the existence of a conjugate linear map, $\phi_V : V^* \rightarrow V$ by $\phi_V(f) = w$, where $V^*$ is a dual space of $V$.