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.


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 $n$.

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

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