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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 bounded(or 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.

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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.

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