$\sup\{||Ax|| : x\in E, A \in \Bbb A\} < \infty$ for every bounded set $E \subset X$ Let $X$ be a Banach Space, $Y$ be NLS and $\Bbb A\subset B(X,Y)$ be such that $\{Ax:A\in \Bbb A\}$ is bounded in $Y$ for every $x\in X$. The $\sup\{||Ax|| : x\in E, A \in \Bbb A\} &lt; \infty$ for every bounded set $E \subset X$.

Why is the problem? IS it not direct consequence from Uniform Boundedness Principle?

$\{Ax:A\in \Bbb A\}$ is bounded in $Y$ for every $x\in X$  means $A$ is pointwise bounded and by UBP $A$ is uniformly bounded.
you can use the UFP and $\exists M,\sup A＜M$.

So $$\sup\{||Ax|| x\in E\}＜M\sup\{||x||,x\in E\}&lt;\infty$$
you can use the UFP and $\exists M,\sup A＜M$.

So $$\sup\{||Ax|| x\in E\}＜M\sup\{||x||,x\in E\}&lt;\infty$$