3
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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 $n$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.