Let $0< \alpha < \frac{1}{2}$ and $C^{\alpha +\epsilon}(0,T, \mathbb{R}^d)$ be the space of Hölder-continuous functions $f:[0,T] \to \mathbb{R}^d$ of exponent $\alpha+ \epsilon$ for $0<\epsilon< \alpha$ and $||\cdot||$ be a norm on $\mathbb{R}^d$. Let us consider the Hölder-norm of order $\alpha$ $$||f||_{\alpha}= \sup_{s \in [0,T]} ||f(s)||+ \sup_{0 \leq s < t <T} \frac{||f(t)-f(s)||}{(t-s)^{\alpha}}$$. If we define the metric $d_{\alpha}(f,g):=||f-g||_{\alpha}$ and fix $\epsilon$, is then the metric space $(C^{\alpha +\epsilon}(0,T, \mathbb{R}^d),d_{\alpha})$ also a complete metric space?
I think its true that $C^{\alpha +\epsilon}(0,T, \mathbb{R}^d) \subset C^{\alpha}(0,T, \mathbb{R}^d)$ but that dont help.