so I understood the whole proof except for one little detail that I couldn't wrap my head around.
for a fixed $y$ ( $φ(x,y) = f(x)$ a linear functional on $H$) by the Riesz representation theorem $\exists ! v \in H \; \text{such that} \; f(x) = \langle x\,,v\rangle \; \forall x \in H$
the author is assuming that $v = Ay$ for some operator $A$
how is that so ?
The author is not assuming that $v = Ay$ for some operator $A$. Quite the contrary, the author is merely using the symbol "$Ay$" to stand in for the vector guaranteed by an application of the Riesz representation theorem to the linear functional $x\mapsto \varphi(x,y)$. The author then goes on to prove that the map $y\mapsto Ay$ is a bounded operator, first by proving linearity, and finally the boundedness.