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Let $T_n$ be a sequence of continuous linear operators from a Banach space $X$ to a normed linear space $Y$. Now, for all $x \in X$, $\lim_{n \rightarrow \infty} T_n(x)$ exists in $Y$. Show that the sequence $T_n$ is uniformly bounded.

I can prove that if we define $T(x) = \lim_{n \rightarrow \infty} T_n(x)$ for each $x \in X$, then $T$ is bounded. The problem is I don't know where to start when proving this proposition.

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You may want to use the principle of uniform boundedness.

Since the limit of $T_n(x)$ exists for all $x\in X$, $\|T_n(x)\|\le C_x$ must be bounded for all $x\in X$ (note that the bound depends on $x$, but not on $n$). The PUB now states that $\|T_n\|$ is bounded.

As Davide Giraudo already noted, the PUB is a consequence of Baires category theorem.

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Use Baire categories theorem with the closed sets $$F_N:=\bigcap_{n\in\Bbb N}\{x,\lVert T_nx\rVert\leqslant N\}.$$

What we prove is the principle of uniform boundedness.

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