## A lemma of J. L. Lions

Let $X,Y$ and $Z$ be three Banach spaces with norms $\|\cdot \|_X,\ \|\cdot \|_Y$ and $\|\cdot \|_Z$. Assume that $X \subset Y$ with compact injection and that $Y\subset Z$ with continuous injection. Prove that

$\forall \varepsilon >0 \exists C_\varepsilon \geq 0$ satisfying $\|u\|_Y \leq \varepsilon \|u\|_X+C_\varepsilon \|u\|_Z,\ \forall u \in X$.

Applications:

1. Prove that for every $\varepsilon >0$ there exists $C_\varepsilon \geq 0$ satisfying$\displaystyle \max_{[0,1]}|u| \leq \varepsilon \max_{[0,1]}|u^\prime|+C_\varepsilon\|u\|L^1,\ \forall u \in C^1([0,1])$.
2. Pick $p>1$. Prove that for every $\varepsilon >0$ there exists $C=C(\varepsilon,p)$ such that $\|u\|_{L^\infty(0,1)} \leq \varepsilon \|u^\prime\|_{L^p(0,1)}+C\|u\|_{L^1(0,1)},\ \forall u \in W^{1,p}(0,1)$.

Source: Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011

Proof: For the initial lemma, just argue by contradiction. The two application are more or less immediate after using the given lemma.