Let $f,g$ be two nonconstant positive functions on $I=[0,1]$ and we assume that : $$ \sup_{x\in I}\sqrt{f^2(x)+g^2(x)}=\sqrt{\sup_{x\in I}f^2(x)+\sup_{x\in I}g^2(x)} $$ This implies that $(f,g)$ are linearly dependent ?
If not what we can deduce on $f$ and $g$ ?
Edit : the first question is false by taking $f=1-x$and $g=1-x^2$.