Lipschitz maps are defined over metric space as maps $f:(X,d\_X) \to (Y,d\_Y)$ such that
$$ d\left( f(x),f(x^\prime) \right)\_Y \le k d(x,x^\prime)\_X \ \forall x,x^\prime \in X, $$
where $k$ is a positive constant. We usually say that $f$ is a *contraction* if $k<1$.


It is well know that a different equivalent metric on $X$ does not preserve contractions, i.e. a map can be a contraction with respect to a metric but not with respect to an equivalent one.


In the Banach space setting, where the spaces $X$ and $Y$ are endowed with a norm defining the topology, there is a somehow "canonical" distance given by
$$ d(x,x^\prime) = \lVert x-x^\prime \rVert .$$
With this distance, Lipschitz maps can be characterized as maps satisfying, for some $k>0$
$$ {\left\lVert f(x) - f(x^\prime) \right\rVert}\_Y \le k {\left \lVert x-x^\prime \right \rVert}\_X \ \forall x, x^\prime \in X. $$
It is obvious that, if $f$ satisfies the above relation, then is a $k$-lipschitz map.




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In the Fréchet space setting, the topology is defined by a countable family of semi-norms $({\lVert\cdot\rVert}\_n)$. The classical example of metric inducing the same topology is given by
$$ d(x,x^\prime) = \sum\_{n=0}^\infty {2^{-n}} \frac{{\lVert x-x^\prime\rVert}\_n}{1+{\lVert x-x^\prime\rVert}\_n} . $$


In analogy with the Banach case, I would like to characterize (at least some) Lipschtiz maps between Fréchet spaces as maps satisfying
$$ {\left\lVert f(x) - f(x^\prime) \right\rVert}\_n \le k {\left \lVert x-x^\prime \right \rVert}\_n \ \forall x, x^\prime \in X,\ \forall n \in \mathbb{N}. $$
Again, maps satisfying the last equation **are** Lipschitz maps with respect to the metric defined above, but the Lipschitz constant is not $k$ anymore, and in particular contraction with respect to the semi-norms (i.e. maps satisfying the last equation with $k<1$) are not contraction with respect to the metric.


Are there equivalent distances on $X$ and $Y$ such that *every* contraction with respect to the semi-norms is a contraction with respect with the new distance? If this is not possibile for *every* contraction, is it possible for a specific one?



Equivalent metrics on Fréchet spaces and Lipschitz maps
Use $\sum 2^{-n}(\|x-y\|\_n \wedge 1)$ for the distance on $Y$ and $\sum 2^{-n}(\|x-y\|\_n \wedge 2)$ for the distance on $X$. 


