I want to show that $||f||=\max_{t\in [0,1]}|f(t)|$ defines a norm on $C([0,1])$.
I only have a question on the triangle inequality property. This is what I have done but I am not quite sure it is right.
Let $f,g\in C([0,1])$, then for all $t\in [0,1]$ we have
$|f(t)+g(t)|\le|f(t)|+|g(t)|\le \max_{t\in [0,1]}|f(t)|+\max_{t\in [0,1]}|g(t)|$
so $\max_{t\in [0,1]}|f(t)|+\max_{t\in [0,1]}|g(t)|$ is an upeer bound for $|f(t)+g(t)$|,
so $\max_{t\in [0,1]}|f(t)+g(t)|\le \max_{t\in [0,1]}|f(t)|+\max_{t\in [0,1]}|g(t)$
thus $||f+g||\le ||f||+||g||$ as desired.
Is this correct?