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.

1 Answers 1

1

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 $n$.

  • 0
    Diverge to infinity.2012-11-27