I'm reading through Real Analysis by Royden & Fitzpatrick and I always try to deal with the first few problems of each section. For the first section on $L^p$ spaces there were mostly routine exercises on proving norms, but this subquestion has me a bit confused:
Show that
a) $\not\exists$ number $c\geq 0$ for which $||f||_{max}\leq c||f||_1$ $\forall f \in C[a,b] $.
b) But $\exists$ $c\geq 0 $ for which $||f||_1\leq c||f||_{max}$ $\forall f \in C[a,b] $
Where:
$||f||_{max}:= max_{x\in[a,b]}|f(x)|$
$||f||_1:=\int_a^b|f|$
EDIT: Does the following counterexample work for part a)?
Let $f(x)=x^n$ on $[0,1]$, $max|f(x)| > c \int_a^b|f(x)|$ leads to $1>|\frac{c}{n+1}|$ and for any c, however large, we can take n=c and that will keep our inequality true.