# all norms are not equivalent

Let $V$ be the vector space of continuous functions $[-1,1]\to\mathbbmss{R}$ that are differentiable at $0$. Then we can define norms

 $\|f\|=\max_{x\in[-1,1]}|f|,$

and

 $\|f\|^{\prime}=\|f\|+|f^{\prime}(0)|.$

It is not difficult to find a sequence of functions $f_{1},f_{2},\ldots$ in $V$ such that

1. 1.

$f_{k}^{\prime}(0)=k$ for $k=1,2,\ldots$,

2. 2.

$\|f_{k}\|=1$.

Then $\|f_{k}\|=1$, and $\|f_{k}\|^{\prime}=1+k$, so there is no $C>1$ such that

 $\|f\|^{\prime}\leq C\|f\|\quad f\in V,$

and $\|\cdot\|$ and $\|\cdot\|^{\prime}$ cannot be .

Title all norms are not equivalent AllNormsAreNotEquivalent 2013-03-22 15:36:11 2013-03-22 15:36:11 matte (1858) matte (1858) 6 matte (1858) Example msc 46B99