Why the following norm of $C[0,1]$ is strictly convex
$||| f|||= ||f||_\infty + ||f||_{L^2 [0,1]}$
where $||f||_{L^2 [0,1]}$ refers to $p$-norm of $L^p[0,1]$ when $p=2$.
I'm not sure if this relevant but I did prove that $||f||_\infty$ isn't strictly convex and $||f||_{L^2 [0,1]}$ is strictly convex.
Thanks
( A norm || || is strictly convex if when ||x||=||y||=1 and ||x+y||=2 so x=y )