Let $(V,\left\Vert \cdot\right\Vert_V),(W,\left\Vert \cdot\right\Vert_W) $ be norm spaces.
For any linear transformation $\;T:V\rightarrow W\;$ we will define:$\;\left\Vert T\right\Vert _{op}=sup\{\left\Vert Tv\right\Vert _{W}:\left\Vert v\right\Vert _{V}=1\}$
Show that the following are equivalent:
$T$ is continuous
$T$ is continuous in $0_v$
$\;\left\Vert T\right\Vert _{op} < \infty $
I want to show this by proving $1 \implies 2 \implies 3 \implies 1 $
$1 \implies 2 \;$ is given.
I would love to get a hint / insight on how to do $ 2 \implies 3 \implies 1 $
Hint. Showing $1\iff 2$ and $1\iff 3$ would be easier.
$2\implies 1$: Use the linearity of $T$: we can see that by translation that the continuity of $T$ at $0$ implies that of any point.
$1\implies 3$: $v\mapsto \|Tv\|$ is continuous, and every continuous function over a closed set has a maximum.
$3\implies 1$: You can show that $\|Tx\|\le\|T\|\|x\|$: Could you see that why $\|Tx/\|x\|\|\le \|T\|$ holds?