Equivalence of norms in $ C_1([0,1])$ Prove or contradict that the following norms are equivalent:
$$
\begin{align*}
\|f\|_1 &= \int_0^1 |f| \; d\lambda+ \max \; |f'|,
\\
\|f\|_2 &=\max \; |f|+\max \; |f'|,
\end{align*}$$
for all $f\in C_1 ([0,1])$.
I did prove it for nonnegative functions (with mean value theorem) but I don't know how to continue from there, I would be glad to get some help.
Thanks!  
 A: Let $f \in C_1([0,1])$
$$\int |f| d\lambda \leq \int \max |f|\, d\lambda, \text{ so }\|f\|_1 \leq \|f\|_2.$$
Now, since $f'$ is continuous on closed interval $[0,1]$ it is bounded. Therefore for each $x \in [0,1]$ we derive
$$
\begin{align}
\|f\|_2 &= \max|f| + \max|f'|\newline
&\leq |f(x)| + \max|f'| + \max|f'|\newline
&\leq |f(x)| + 2\|f\|_1 - 2\int |f|d\lambda\newline
&\leq |f(x)| + 2\|f\|_1.
\end{align}
$$
However we know that $\exists_{x_0} |f(x_0)| \leq \int |f|\, d\lambda$, so we have $|f(x_0)| \leq \|f\|_1$ and finally
$$\|f_2\| \leq 3\|f\|_1.$$
Why $\max|f| \leq |f(x)| + \max |f'|$?
Let $x,y \in [0,1]$. Then $\exists_{c \in [0,1]}$ such that $$\big| |f(x)| - |f(y)|\big| \leq |f(x) - f(y)| \leq |f'(c)| \leq \max |f'|$$ by the mean value theorem.
A: Obviously $\|f\|_1 \leq \| f \|_2$. This tells us that the identity operator $(C^1[0,1],\| \cdot \|_2) \to (C^1[0,1], \| \cdot \|_1 )$ is continuous, if we can prove this is an homeomorphism we're done. We know that it's a bijective, bounded linear map so, by the open mapping theorem, it suffices to check that $\| \cdot \|_1$ induces a complete metric. 
Take a Cauchy sequence in $(C^1[0,1], \| \cdot \|_1 )$ then, in particular, $f_n'$ is a Cauchy sequence in $C[0,1]$ so $f_n'\to g\in C[0,1]$ uniformly. Since $f_n$ is Cauchy in $L^1$ we have $f_n \to f\in L^1[0,1]$. It's standard that there exists a subsequence of the $f_n$ that converges pointwise a.e. in particular it does so at a point, but the sequence of derivatives converges uniformly, so that $f\in C^1[0,1]$ and $g=f'$.  
