Operator norm and equivalent definitions

Question

From the definition of the operator norm, we have:

$||T||_{op}=\inf\{c\in \mathbb{R}^+:||Tv||\leq c||v||, v \in V\}$

If $T: V \rightarrow W$ is a linear map between two normed vector spaces. I have the following problem, if we have (for $\epsilon, k$ positive constants):

$x \in V, ||x||_{V} < \epsilon \Rightarrow ||T(x)||_{W} \leq k$

Then how does it follow that $||T||_{op} \leq k\epsilon^{-1}$ using the above definition? I tried using an alternative definition (namely the $\sup$ definition for $||v|| \leq 1$ and it works), but for this definition I'm not sure how to prove the result.

Thanks for your help.

Answer 1

You can write $$ \|T\|_{\rm op}=\inf\{c>0: \|Tv\|\leq c,\ v\in V,\ \|v\|\leq1\}. $$ If $\|x\|<1$, then $\|\epsilon\,x\|<\epsilon$, and so $\|T(\epsilon x)\|\leq k$, i.e. $$ \|Tx\|\leq\,\frac k\epsilon. $$ Thus, $$ \|T\|_{\rm op}\leq\frac k\epsilon. $$ Technically, you still need to adjust for $\|x\|\leq1$ instead of $\|x\|<1$, but that is not a big deal.