Normed Linear Space - maximum norm vs. $||f||_1$ For $f$ in $C[a,b]$ define $$|| f ||_1 =\int_a^b |f|.$$ a. Show that this is a norm on $C[a,b]$. b. Show that there is no number $c \geq0$ for which $$||f||_{max} \leq c ||f||_1 \ for \ all \ f \ in \ C[a,b]$$ c. Show there is a $c\geq 0$ for which $$||f||_1 \leq c||f||_{max} \ for \ all \ f \ in \ C[a,b]$$ (where $||f||_{max}=\max\limits_{x \in [a,b]} |f(x)|)$ I proved a. by showing all norm properties (the triangle inequality, positive homogeneity and nonnegativity). I also proved part b. by false assumption that leaded me to contradiction Any help on part c. Thanks

It holds that $|f(x)| \leq \max_{x \in [a,b]} f(x):=||f||_{\max}, \forall x \in [a,b]$.

Thus:

$$||f||_1= \int_a^b |f(x)| dx \leq \int_a^b ||f||_{\max} dx= ||f||_{\max} \int_a^b 1 dx=(b-a) ||f||_{\max}$$