Since my reputation is too low I could not add a comment. However, I recently met with this problem when I was reading Van der Vaart& Wellner (1996). According to this book, a hidden assumption for @angryavian's answer to hold is that the norm must satisfy the Riesz property ($L_r(Q)$-norms do possess this property), i.e., $$ |f| \le |g| \Rightarrow \|f\| \le \|g\|. $$ so that $$ \begin{aligned} |f - l| \le |u - l| &\Rightarrow \|f - l\| \le \|u - l\| \\ |f - u| \le |u - l| &\Rightarrow \|f - u\| \le \|u - l\| \end{aligned} $$

I hope this could serve as a complement.

Since my reputation is too low I could not add a comment. However, I recently met with this problem when I was reading Van der Vaart& Wellner (1996). According to this book, a hidden assumption for @angryavian's answer to hold is that the norm must satisfy the Riesz property ($L_r(Q)$-norms do possess this property), i.e., $$ |f| \le |g| \Rightarrow \|f\| \le \|g\|. $$ so that $$ \begin{aligned} |f - l| \le |u - l| &\Rightarrow \|f - l\| \le \|u - l\| \\ |f - u| \le |u - l| &\Rightarrow \|f - u\| \le \|u - l\| \end{aligned} $$

I hope this could serve as a supplement.