I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $X,Y,Z$ be real Banach spaces with corresponding norms $|\cdot|_X, |\cdot|_Y, |\cdot|_Z$. Assume that $X \subset Y$ with compact injection and that $Y \subset Z$ with continuous injection.
- Prove that for every $\varepsilon>0$ there is $C_\varepsilon > 0$ such that $$ |u|_Y \le \varepsilon |u|_X + C_\varepsilon |u|_Z \quad \forall u \in X. $$
- Prove that for every $\varepsilon>0$ there is $C_\varepsilon > 0$ such that $$ \|u\|_\infty \le \varepsilon \|u'\|_\infty + C_\varepsilon \|u\|_{L^1} \quad \forall u \in C^1([0, 1]). $$
There are possibly subtle mistakes that I could not recognize in below attempt of (2). Could you please have a check on it?
Let $X:=C^1([0, 1]), Y :=C([0, 1])$ and $Z:=L^1([0, 1])$. Let $|\cdot|_X$ be the $C^1$-norm, $|\cdot|_Y$ the supremum norm and $|\cdot|_Z$ the $L^1$-norm. Then we have $|u|_Z \le |u|_Y$ and $|u|_X=|u|_Y+|u'|_Y$ for $u \in X$. Then $X \subset Y$ and $Y \subset Z$ with continuous injections.
Let's prove that $X \subset Y$ with compact injection. Let $B_X$ be the closed unit ball of $X$, i.e., $$ B_X := \{u \in X : |u|_X \le 1\} = \{u \in X : \|u\|_\infty + \|u'\|_\infty \le 1\}. $$
For $u \in B_X$, we have $$ |u(x)-u(y)| \le \|u'\|_\infty |x-y| \le |x-y| \quad \forall x, y\in [0, 1]. $$
Then $B_X$ is uniformly equi-continuous. Clearly, $B_X$ is uniformly bounded in $Y$. Then $B_X$ has compact closure in $Y$ by Arzelà–Ascoli theorem.
By (1), for every $\varepsilon>0$ there is $C_\varepsilon > 0$ such that $$ \|u\|_\infty \le \varepsilon (\|u\|_\infty + \|u'\|_\infty) + C_\varepsilon \|u\|_{L^1} \quad \forall u \in X. $$
Then for every $\varepsilon>0$ there is $C_\varepsilon > 0$ such that $$ \|u\|_\infty \le \frac{\varepsilon}{1-\varepsilon} \|u'\|_\infty + \frac{C_\varepsilon}{1-\varepsilon} \|u\|_{L^1} \quad \forall u \in X. $$
Clearly, $\frac{\varepsilon}{1-\varepsilon}$ can be arbitrarily small. The claim then follows.
