# Equivalence of p-norms in $C^{0}_{p}[a,b]$

1. Jan 12, 2012

### ELESSAR TELKONT

1. The problem statement, all variables and given/known data

Let $1\leq s<r<\infty$. For what pairs $s,r$ the norms $\left\|\cdot\right\|_{s},\left\|\cdot\right\|_{r}$ are equivalent?

2. Relevant equations

I have already proven that $\left\|\cdot\right\|_{s}\leq (b-a)^{\frac{r-s}{rs}}\left\|\cdot\right\|_{r}$ and that $\left\|\cdot\right\|_{s}\leq (b-a)^{\frac{1}{s}}\left\|\cdot\right\|_{\infty}$. Of course I have proven, exposing that
$x_{k}(t)=\begin{cases} 1-kt &0\leq t\leq\frac{1}{k}\\ 0 &\frac{1}{k}\leq t\leq 1 \end{cases}$
has a finite infinite-norm for all k, but the 1-norm goes to 0 as k goes infinity, then I can say that there's no c>0 that $\left\|\cdot\right\|_{\infty}\leq c\left\|\cdot\right\|_{s}$, and of course this is true for $[a,b]$ since I can map the [0,1] to the [a,b].

3. The attempt at a solution

I can't imagine some continuous function sequence that may be used as a counterexample to say that I can't bound the s-norms with the r-norms from below. I have done the work as I have proven for the norms in lp sequence spaces and I get that they are equivalent but I suspect they aren't. Please help.