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I have a short question about the proof found here: http://mathonline.wikidot.com/the-uniform-boundedness-principle

It shows that given an $x$ s.t its norm is $1$., then the linear functionals is bounded. However, I do not see how it implies for any $x$, $\|T(x)\|\leq M \|x\|$ Whats the trick to extend the boundedness to all x?

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$$\|T(x)\| = \|T(\frac{x}{\|x\|})\|x\|\| = \|T(\frac{x}{\|x\|})\| \cdot \|x\| \leq M\|x\|$$ for any $x$, where we used the linearity of $T$ and the homogeneity of the norm.

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