Holder continuous H-valued function

Question

It is proven that the space of real-valued Holder continuous function $C^\alpha[0,T]$ is compactly embedded in $C^\beta[0,T]$, where $\alpha>\beta$.

Does this hold for $C^\alpha([0,T];H)$ and $C^\beta([0,T];H)$ where $H$ is a Hilbert space?

Answer 1

I think not (if $H$ is infinite-dimensional). Let $(e_1,e_2,\dots)$ be orthonormal in $H$ and define the constant functions $$f_n(t)=e_n,\quad\forall t\in[0,T].$$ Then clearly $\|f_n\|_{C^{\alpha}}\equiv 1$ so that $\{f_n\}$ is a bounded set in $C^\alpha$, but $\|f_n-f_m\|_\infty=2$ if $n\neq m$, so no subsequence will converge uniformly let alone in $C^\beta$.