$h$ is a Lipschitz function, of order $\alpha\in (0,1]$, for $s<1$, $T(h)=\frac{1}{s}\sum\limits_{k=1}^{n} a_k(h\circ f_k)(x)$, could anyone help me to show $T$ is lipschitz map, where $f_k$ are Lipschitz, bounded map for all $k$, $0<a_k<1$. $d$ is the metric and they are all in $X$ a seperable metric space. Thanks for the help. I understand $T$ is a linear map, so continous already, right?

  • $\begingroup$ "h is a Lipschitz function, of order α". This is your first sentence. Do you mean that h is a Holder function, of order α? $\endgroup$ – Holding Arthur Apr 15 at 12:50
  • $\begingroup$ I mean: $d(h(x), h(y)\le d(x,y)^{\alpha}\quad \forall x,y$ $\endgroup$ – Ding Dong Apr 15 at 12:51
  • $\begingroup$ Are you sure you are not missing a constant mutiple? $\endgroup$ – Holding Arthur Apr 15 at 13:58
  • $\begingroup$ OKay, you can put some $C$ before $d(x,y)^{\alpha}$ $\endgroup$ – Ding Dong Apr 15 at 14:59
  • $\begingroup$ Is $X$ a linear normed space? Could you provide some context? $\endgroup$ – Holding Arthur Apr 15 at 22:03

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