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.