Referring to this question: https://math.stackexchange.com/q/1355854 which was in turn a question about: $a: E\times F\to G$ bilinear separately continuous implies continuous?
I don't understand where $$\left\|\dfrac{a_x}{\|x\|}\right\|≤c$$ comes from, still.
The theorem merely states that $\|a_x y\|\leq c\|y\|$ for all $x\in E$, $y\in F$.
Let me write it out more explicitly. Define a linear mapping $T : E \to \mathcal B(F, G)$ by $$ x \mapsto a_x,$$ where $a_x \in \mathcal B(F, G)$ is defined by $$ a_x(y) = a(x,y).$$ Notice that $a_x$ is indeed in $\mathcal B(F, G)$, since the mapping $y \mapsto a(x,y)$ is continuous for each $x \in E$.
Let $B(1) \subset E$ be the unit ball in $E$. We are going to show that $T(B(1))$ is a bounded subset of $\mathcal B(F, G)$.
By the uniform boundedness principle, it is only necessary to show that for every $y \in F$, the set $$\{ T(x)( y ) : x \in B(1) \} = \{ a(x,y) : || x || \leq 1 \}$$ is a bounded subset in $G$. This statement is true, because the mapping $x \mapsto a(x,y)$ is continuous for every $y \in F$.
Having shown that $T(B(1))$ is bounded, we deduce that there exists a $c$ such that $$ || x || \leq 1 \implies ||a_x || \leq c.$$ Therefore, $$ || a_x || \leq c ||x ||$$ for all $x \in E$.
Finally, $$ || a(x,y) || \leq ||a_x || ||y || \leq c || x || \ || y ||.$$