The following result is a corollary of the principle of uniform boundedness.

Theorem - Let $X$ be a Banach space and $Y$ a normed vector space. Let $(T_n) \in B(X,Y)$ be a sequence of bounded operators from $X$ to $Y$ . If $(T_nx)$ converges for every $x \in X$ , then the operator

Proof : It is clear that the operator $T$ is linear.

For each $x \in X$ we have $\displaystyle \;\sup_n \|T_nx\| < \infty\;$ since $(T_nx)$ is convergent. By the principle of uniform boundedness we must also have $\displaystyle M := \sup_n \|T_n\| < \infty$ .

Then for each $x \in X$ we have

which means that $T$ is bounded. $\square$