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Let $T:X \to Y $be a linear operator between two normed spaces X and Y. Show that if for any sequence ${x_{n}}\subseteq X$, ${x_{n}} \to 0$ implies $Tx_{n}$ is bounded,then T is bounded.

My solution: By contraposition if T is not bounded, then $Tx_{n}$ is not bounded implies for some sequence ${x_{n}}\subseteq X$, ${x_{n}} \to 0$ .

We know that if T is not bounded,then for all $\epsilon>0, \sup||Tx||=\infty$

I dont know how proceed solution help me please?

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Suppose $T$ is not bounded. Then there exists a sequence $(y_n)$ such that

$$ \frac{\|Ty_n\|}{\|y_n\|} \to \infty \text{ as } n\to\infty $$

Take $$ x_n = \frac{y_n}{ \|y_n\| \|Ty_n\|} $$

Then $x_n \to 0$ as $n \to \infty$ and

$$ \frac{\|Tx_n\|}{\|x_n\|} = \frac{\|Ty_n\|}{ \|y_n\| \|Ty_n\|} \cdot \left(\frac{\|y_n\|}{ \|y_n\| \|Ty_n\|}\right)^{-1} = \frac{\|Ty_n\|}{\|y_n\|} \to \infty \text{ as } n\to\infty $$

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