2
$\begingroup$

Consider two Frechet spaces $(X,d_X),\;(Y,d_Y)$. Let $T:X \rightarrow Y$ be a continues linear operator. Is it true that $T$ is Lipschitz in the following sense: $$\exists \alpha > 0 \; \forall x,y \in X \; d_Y(Tx,Ty)\leq \alpha d_X(x,y)$$

By the definition of continuity and translation invariance of the metric we know there exists $\delta > 0$ such that $$ d_X(x,y)<\delta \implies d_Y(Tx,Ty)<1$$ But I am not sure whether this imples the former condition.

$\endgroup$
1
$\begingroup$

Suppose $X$ is a normed linear space and $Y=X$ with the metric $d_Y(x,y)=\frac {\|x-y\|} {1+\|x-y\|}$. Then $Y$ is a Frechet space. Let $T:(Y,d_Y) \rightarrow (X,\Vert\cdot\Vert)$ be the idenitity map. $T$ is continuous. If the inequality you have stated holds then $\|x-y\| \leq \alpha$ for all $x,y$, which is false.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.