As a part of a proof I stuck at an inequality. Suppose $u$, $v$, and $w$ are functions in $L^2$ space,
$$
(\|v\| + \|v'\|)(\|w\|+\|w'\|) \le \sqrt{2} (\|v\|^2+\|v'\|^2)^{1/2}\sqrt{2}(\|w\|^2+\|w'\|^2) ^{1/2}
$$
Considering the norm is in $L^2$. How did those $\sqrt{2}$ appear?