# Equivalence of two norms

Define two norms as following: $$\left\Vert f\right\Vert _{1}={ \max_{0\leq x\leq1}\left|f\left(x\right)\right|} , \quad\text{ and }\quad \left\Vert f\right\Vert _{2}={\intop_{0}^{1}\left|f\left(x\right)\right|dx}$$

on the vector space $C\left[0,1\right]$ (the continuous functions).

I need to prove that the two norms aren't equivalent.

## 1 Answer

Let $f_n(x):=\max\{1-nx,0\}$; it's a continuous function for all $n$. Its $1$ norm is $1$, but its $²$-norm explodes with $n$.

• what do you mean explodes with n? Im not familer with this term. – sony jimbo Nov 27 '12 at 13:49
• Diverge to infinity. – Davide Giraudo Nov 27 '12 at 15:43