Let $X$ be a Banach's space $Y$ a normed vector space $H\in B(X,Y)$ a family of bounded linear transformations $X\rightarrow Y$ and $V_n:=\{x\in X:\exists T\in H$ such that $||Tx||>n \} n\in \mathbb{N}$
If $\exists N\in \mathbb{N}$ such that $V_N$ isn't dense then $\sup_{T\in H}||T||<\infty$
I'm not sure how to relate density with $T$'s finiteness, I had the idea that $T$ being bounded gave $$\sup_{T\in H}||T||<\infty$$ automatically.