Suppose we have a decreasing family of Banach spaces $H^s$ (the prototype is some variant of the Sobolev space $H^s=W^{s,2}$, so you can assume that they are Hilbert spaces if you want), and we consider the residual space: $H^{\infty}=\cap_{s>0}H^s$ (for sure, the intersection can be taken over any sequence of $s$ going to $\infty$).
Now $H^\infty$ does not necessarily have a norm to induce its topology, which is induced from the topologies of $H^s$.
And we have a map $T$ from $H^s$ to $H^{s-1}$ satisfy some contraction condition: $ |T(x)-T(y)|_{H^{s-1}} \leq k_s|x-y|_{H^s} $, with $k_s <1$. (Hence $T$ sends $H^\infty$ into itself).
Is there any fixed point theorem for the map $T$ restricted to $H^{\infty}$?
(i) What if we assume $k_s$ depend on $s$?
(ii) What if we can make $k_s$ to be uniformly smaller than 1?
Notes: $T$ does not have to be linear. Here we assume that $|u|_{H^{s_1}} \leq |u|_{H^{s_2}}$ for $s_1\leq s_2$.