Let $H$ be a Hilbert space and let $H^*$ be the dual space of $H$.
The Riesz lemma states that for each $T\in H^*$, there is a unique $y_T\in H$ such that $T(x)=(y_T,x)$ $\forall x\in H$. Also, $||y_T||_H$ = $||T||_{H^*}$.
This doesn't show that there is a one-to-one correspondence between the hilbert space $H$ and its dual space $H^*$.
So, to derive the one-to-one correspondence between $H$ and $H^*$, we need to show that for each $y\in H$, then there exists an unique $T_y \in H$ such that $T_y(x)=(y,x)$ $\forall x \in H$.
I was thinking about showing that $T_y(x)=(y,x)$ for some subspace(s) (whose span is $H$). But I wasn't successful in doing this. I'm also thinking about the Cauchy-Schwarz inequality to show an equality (by showing an inequality in both directions...)