On Continuously Differentiable Vector-Valued Functions of Non-Integer Order

The function spaces of continuously differentiable functions are extensively studied and appear in various mathematical settings. In this context, we investigate the spaces of continuously fractional differentiable functions of order $\alpha >0$, considering both the Riemann-Liouville and Caputo fractional derivatives. We explore several fundamental properties of these spaces and, inspired by a result of Hardy and Littlewood, we compare them with the space of Hölder continuous functions. Our main objective is to establish a rigorous theoretical framework to support the study and further advancement of this subject.