# Difference between a bounded linear map and a Lipschitz linear map

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?

• There is no difference. – Kavi Rama Murthy Dec 17 '18 at 8:24
• No. In the case of linear operators, those definition coincide. More interesting is the case of of nonlinear operators. – Jonas Dec 17 '18 at 8:24

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