Consider on $\mathcal{C}([0,1])$ the following two norms:
$$
\|f\|_{\infty}=\sup _{x \in[0,1]}|f(x)|, \quad\|f\|_1=\int_0^1|f(x)| d x .
$$
Show that $\|\cdot\|_{\infty}$ and $\|\cdot\|_1$ are not equivalent norms.
Hint: Consider the sequence $f_n(x)=x^n, n \in \mathbb{N}$, and show that the equivalence of norms fails on this sequence.
Buongiorno, nello svolgere questo esercizio non capisco come si faccia..
Ringrazio chiunque risponderà!