Fix $p \geq 1$, let $W([0,1])$ be the space of absolutely continuous functions such that for all $f \in W$ we have $\|f'\|_p^p <\infty$. Then this is a Banach Space with the norm; $\|f\|_W=\left(\|f\|_p^p+\|f'\|_p^p\right)^{1/p}$. I want to show that $\iota:W \hookrightarrow C[0,1]$, the inclusion into the space of all continous functions with infinity norm, is bounded. So I have to write;
$$\|f\|_{\sup}\leq c\|f\|_W$$ for some constant $c$ yeah? But I can't seem to summon the technical skill to relate these norms... The only thing I tried with any promise was;
$$\|f\|_{\sup}=\sup_{x\in [0,1]}|f(x)-f(0)+f(0)|\leq \sup_{x\in [0,1]}\left|\int_0^xf'(t)dt\right|^{p/p}+|f(0)|$$ $$\leq \sup_{x\in [0,1]}\left(\int_0^x|f'(t)|^pdt\right)^{1/p}+|f(0)|\leq \|f'\|_p+|f(0)|$$
The question is in a chapter regarding compact operators (and the next question is to show $\iota$ is compact for p>1) but I can't see how any of the material can be relevant here?
Use the estimate $$|f(x)| \le |f(x) - f(y)| + |f(y)| \le \int_{[x,y]} |f'(s)| ds + |f(y)| \le \int_{[0,1]} |f'(s)| ds + |f(y)|\, .$$
This is true for any $y$, so integrate with respect to $y$ to obtain $$ |f(x)| \le \int_0^1 |f'(s)| ds + \int_0^1 |f(s)| ds \, . $$ Use Hoelder's inequality to go to $p$ norms, take the supremum with respect to $x$, and you are done. You can easily trace the constants.