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I would appreciate it if you could give me a few hints as to how I should go about solving this problem.

Suppose $T:\mathfrak{X}\rightarrow \mathfrak{Y}$ is bounded. Show that $\left\|T\right\|=\inf\{K\in\mathbb{R}:\left\|Tx\right\|_{\mathfrak{Y}}\leq K\left\|x\right\|_{\mathfrak{X}} \}$.

I know that $T$ is bounded if $\left\|T\right\|:=\sup\{\left\|Tx\right\|_{\mathfrak{Y}}:\left\|x\right\|_{\mathfrak{X}}\leq 1 \}$. How can I go from $\sup$ to $\inf$?

Thanks in advance

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