Currently I'm self studying functional analysis, namely adjoint operators. In the text, the author gives the following construction/definition of adjoint operators for bounded operators between Hilbert spaces:
Construction/definition: In the case of Hilbert spaces for the operator $A\in L(\mathcal{H}_1\to\mathcal{H}_2)^{[1]}$ the adjoint operator $A^*\in L(\mathcal{H}_2\to\mathcal{H}_1)$ is defined via the relation $$ \langle Ax,y \rangle=\langle x,A^*y \rangle, \text{ for }x\in\mathcal{H}_1\text{ and }y\in\mathcal{H}_2. $$
Prior to this construction/definition, the author wrote the following paragraph:
Paragraph: Applying this notation to the pairing between $X$ and $X^*$, we can rewrite the definition of the dual operator $A^*$ of an operator $A$ is the following (symmetric) way: for all $x\in X$ and for all $f\in Y^*$, $$ \langle Ax,f \rangle=\langle x,A^*f \rangle. $$
My question is fairly straightforward: why in the Construction/definition did the author not take $y\in\mathcal{H}^*_2$, but in Paragraph the author required that $f\in Y^*$? I have a feeling the Riesz representation theorem is hiding in here somewhere, but I don't see this construction fully. Any help is appreciated!
$^{[1]}$ Note here that $L(X\to Y)$ is just notation for the space of all (continuous) bounded operators between normed spaces $X$ and $Y$.