Given is the definition $$C^0([a, b]): ||f|| = \sqrt{|f|} = \sqrt{<f|f>} = \sqrt{\int_a^b{f(x)^2dx} }$$
As an example, let's assume that $f(x) = x, x \in [0, 1]$. Therefore
$$||f|| = \sqrt{\int_0^1{x^2dx}} = \sqrt{\frac{1}{3}} = 0.577$$
From my understanding, in this case the norm signifies the maximum value of the distance of $f$ to 0 for the given interval, but by looking at the graph of $f(x)$ you'd easily see that it's 1 and not 0.577. So what exactly does the norm mean in this case?
Thanks!