# certain transformation of Lipschitz maps

$$h$$ 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 $$k$$, $$0. $$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?

• "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? – Holding Arthur Apr 15 at 12:50
• I mean: $d(h(x), h(y)\le d(x,y)^{\alpha}\quad \forall x,y$ – Ding Dong Apr 15 at 12:51
• Are you sure you are not missing a constant mutiple? – Holding Arthur Apr 15 at 13:58
• OKay, you can put some $C$ before $d(x,y)^{\alpha}$ – Ding Dong Apr 15 at 14:59
• Is $X$ a linear normed space? Could you provide some context? – Holding Arthur Apr 15 at 22:03