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Looking at this problem:

Let $\lbrace T_{n} \rbrace_{n=1}^{\infty}$ be a sequence in $\mathcal{L}(X,Y)$, where $X$ and $Y$ are Banach spaces. Prove that the sequence $\lbrace ||T_{n}|| \rbrace_{n=1}^{\infty}$ is bounded if and only if for each $x \in X$, the sequence $\lbrace T_{n}(x) \rbrace_{n=1}^{\infty}$ is bounded in Y.

I'm not exactly sure how to even start it. I suspect that I need to use uniform boundedness, but I cant see exactly how it fits. Any help is appreciated!

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This a straightforward application of the uniform boundedness theorem.

Suppose that for every $x$ the sequence $T_n(x)$ is bounded, the uniform boundedness theorem implies that the sequence $\|T_n\|$ is bounded.

On the other hand, suppose that the sequence $\|T_n\|$ is bounded, there exists $C$ such that for every $n$, $\|T_n\|<C$, we deduce that $\|T_n(x)\|\leq C\|x\|$ for every $n$.

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