A norm in the $C^1$ space

Question

How do I prove that a norm defined by $$\left\Vert f\right\Vert =\left|f(0)\right|+\int_{0}^{1}\left|f'(x)\right|dx$$ is a norm in the $C^1[0,1]$ for functions of class $C^1$?

Answer 1

All axioms are easy to verify except this one:$$\vert\vert f \vert\vert=0 \Longleftrightarrow f=0$$

Assume $\vert\vert f \vert\vert=0$. Then $f(0)=0$ and $\int_0^1 \vert f'(x)\vert dx=0$

Take $g(x)=\vert f'(x) \vert$ and note that $g$ is non-negative and $\int_0^1 g(x) dx=0$. Suppose $g \neq 0$ . Now use continuity of $g$ to prove there exist an interval $[a,b] \subset [0,1]$ such that $g(x) >0 ,\;\forall x \in [a,b]$.

Now, $$\int_0^1 g(x) dx\geq\int_a^b g(x) dx>0$$ a contradiction! so $g$ must be a zero function! Hence $f'(x)=0$ and so $f$ is constant. But $f(o)=0$ implies $f$ is a zero function