Exercise 4, § 2

Let $\alpha \in ]0,1]$. Let I be a compact interval of $\mathbf{R}$. We denote by $\mathscr{C}^{\alpha}(I)$ the vector subspace of $\mathscr{C}(I)$ formed by the functions $f$ such that there exists $C\geqslant 0$ satisfying

for all $(x, y)\in I\times I$.

We equip $\mathscr{C}^{\alpha}(I)$ with the norm

$\|f\|_{\alpha}=$ sup$_{x\in I}|f(x)|+$ sup$_{(x,y)\in I\times I}\frac{|f(x) -f(y)|}{|x-y|^{\alpha}}$.

$x\not =y$

(a) The space $\mathscr{C}^{\alpha}(I)$ is a Banach space.

(b) The canonical injection $\mathscr{C}^{\alpha}(I)\rightarrow \mathscr{C}(I)$ is compact.

(c) If $0< \alpha < \beta \leqslant 1$, then the canonical injection $\mathscr{C}^{\beta}(I)\rightarrow \mathscr{C}^{\alpha}(I)$ is compact.

(d) The Sobolev space $H^1(\mathring{I}) ($cf. IV, p. 221, n$^o14$) identifies with a subspace of $\mathscr{C}^{1/2}$(I), but the canonical injection $H^1(\mathring{I})\rightarrow \mathscr{C}^{1/2}(I)$ is not compact.