Banach-Steinhaus theorem

Let $X$ be a Banach space and $Y$ a normed space.If a family $ℱ\subset ℬ(X,Y)$ of bounded operators from $X$ to $Y$ satisfies

for each $x\in X$, then

i.e. $ℱ$ is a bounded subset of $ℬ(X,Y)$with the usual operator norm. In other words,there exists a constant $c$ such that for all $x\in X$ and $T\in ℱ$,

$$\parallel Tx\parallel \le c\parallel x\parallel .$$