Reading this answer here, I didn't understand the last $\color{red}{\leq}$ in: $$\Vert a(x,y)\Vert_G=\Vert a_x(y)\Vert_G\leq\Vert a_x\Vert\Vert y\Vert_F\color{red}{\leq} c\Vert x\Vert_E\Vert y\Vert_F,$$since we only had $\|a_x\| \leq c$, from the start. I left a comment there, but then I saw the answer is $2$ years old, so I'm not expecting feedback on that side.
Can someone clarify this for me? Thanks.
By uniform boundedness principle, there is a $c$ such that for all $x\in E$, $\left\|\dfrac{a_x}{\|x\|}\right\|≤c$. So $∥a_x∥≤c\|x\|_E$.