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 $(T_{n}x)$ converges for every $x\in X$, then the operator

$T:X\longrightarrow Y$

is linear and bounded. Moreover, the sequence $(\|T_{n}\|)$ is bounded.

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

For each $x\in X$ we have $\displaystyle\;\sup_{n}\|T_{n}x\|<\infty\;$ since $(T_{n}x)$ 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. $\square$