Difference between a bounded linear map and a Lipschitz linear map

Question

Let $X$ and $Y$ be normed linear spaces over a scalar field, and let $T:X\to Y$ be a linear map. Then, $T$ is said to be bounded if there exists some constant $K\geq 0,$ such that \begin{align} \Vert T(x) \Vert \leq K\Vert x\Vert ,\;\;\forall\;x\in X. \end{align}

In my book here, the same definition is being given to $T$, being Lipschitz.

So, my question is: Is there any difference between a bounded linear map and a Lipschitz linear map?

Answer 1

No, there is no difference. If $T:X\to Y$ is linear, then

$T$ is bounded $ \iff T$ is Lipschitz.