# Uniform Boundedness Principle question about the proof.

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?

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