The Reisz' Lemma reads
Let $H$ be a hilbert space over a field $\mathbb{K}$. Then,$$\forall f \in H^{*} \quad \exists! y \in H: f(x)=\langle x,y \rangle \, \forall x \in H , \|f\|=\|y\|$$ Now, let $\dim H < \infty$. I know that one can show that $H$ and its dual space is isomorphic quite easily, but I want to see how this is a consequence of this lemma. I think that this lemma gives you a map from the dual space of $H$ to $H$ given by $f \mapsto y$, but I want to show it is injective and surjective. Does this follow from the fact that this is an isometric map, and if so, how?
Also, I see textbooks defining the dual space basis $\theta_i$ from the basis of $H$ that is $e_i$ by $\langle e_i,\theta_i \rangle=\delta_{ij}$. Is this an abuse of notation?