To Prove - $C[0,1], 0<p<1$ (with usual p-norm) is not a norm linear space.
My thought process - I know that the triangle inequality is violated. But I am not able to produce a counterexample because every time I try to construct an example, I violate that it must be continuous on $[0,1]$. One more subtle point, most other similar proofs that I've done usually are with the $f = g$(or $x = y$ for the two points of Triangle Inequality). But here, taking $f = g$ will not provide a counterexample since for such a case, Triangle Inequality holds with equality.
To prove/disprove - $C[0,1], $p = inifinity$ $ (i.e. the norm defined the supremum of the absolute value of the function on the closed interval $[0,1]$.) is not a norm linear space.
My thought process - The proof for p>=1 uses Holder Inequality, Minkowski Inequality. I wish to know if that thing can simply be used for p = infinity also. Then, that's done.