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**dori1123** Let $\displaystyle F$ be a collection of continuous linear maps from a Banach space $\displaystyle X$ into a normed space $\displaystyle Y$. Then the following are equivalent:

(1) $\displaystyle sup_{T \in F}||Tx||< \infty \forall x \in X$

(2) there exists $\displaystyle t>0$ such that $\displaystyle ||Tx|| \leq t \forall x \in X$ with $\displaystyle ||x|| \leq 1$ and $\displaystyle T \in F$

(3) there exists $\displaystyle t>0$ such that $\displaystyle ||Tx|| \leq t||x|| \forall x \in X$ and $\displaystyle T \in F$