Consider an operarator $A: D(A) \subset X \to X$ generating an analytic semigroup $e^{At}$ of negative type and define as usual (for example see [Pazy]) the negative powers $(-A)^\theta$ for $0< \theta < 1$.
Consider the following set
$$K_M=\{u \in C([0,T];H) : |u|_{C^\beta([0,T];H)}+ \sup_{t\leq T} |u(t)|_{D(-A)^\theta} \leq M\}$$ Then Cerrai in [Cerrai, A KHASMINSKII TYPE AVERAGING PRINCIPLE FOR STOCHASTIC REACTION–DIFFUSION EQUATIONS, corollary 4.5] says that this is a compact set and ok by Ascoli Arzelà this is certainly true.
My question is: is the following compact? $$K^2_M=\{u \in C([0,T];H) : |u|_{C^\beta([0,T];H)}+ \sup_{t\leq T} |u(t)|_H \leq M\}$$
No, this set is not compact (unless $H$ is finite-dimensional or $M=0$). For $\xi\in H$ with $|\xi|_H\leq M$ let $u_\xi\colon [0,T]\to H,\,t\mapsto \xi$. Then $u_\xi\in K_M^2$ and $\xi\mapsto u_\xi$ is an isometry from $\{\xi\in H:|\xi|_H\leq M\}$ into $K_M^2$. But the first set is not compact.