Bounded inverse (Brezis)

Question

In the proof of the corollary 2.7 (T bounded, bijective so $T^{-1}$ bounded) the autor uses the conclusion of the open map theorem

$TB_E(0,1)\supset B_F(0,c)$

and the injectivity of the Operator to conclude that

if $x\in E$ is chosen in such a way that $||Tx||\leq c$ then $|x|\leq 1$

and by homogeneity it concludes that

$|x|\leq \frac 1 c ||Tx||$ for all $x \in E $

I cant understand how he find the last inequality

Answer 1

Take some $x\ne0$ then $Tx\ne0$. Set $y:=\frac{c}{\|Tx\|}x$. Then $\|Ty\|=c$, which implies $\|y\|\le 1$, and $ \|x\|\le c^{-1} \|Tx\|$.