Existence of a conjugate linear map

Question

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 $V$.

Answer 1

The existence of $\phi_v$ is a direct consequence of the theorem of representation you mention.

The only thing to prove is that $\phi_V$ is linear. And this is immediate using the uniqueness of representation as stated by theorem 6.8 you mention.