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

  1. Jan 12, 2012 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    I have already proven that [itex]\left\|\cdot\right\|_{s}\leq (b-a)^{\frac{r-s}{rs}}\left\|\cdot\right\|_{r}[/itex] and that [itex]\left\|\cdot\right\|_{s}\leq (b-a)^{\frac{1}{s}}\left\|\cdot\right\|_{\infty}[/itex]. Of course I have proven, exposing that
    [itex]
    x_{k}(t)=\begin{cases}
    1-kt &0\leq t\leq\frac{1}{k}\\
    0 &\frac{1}{k}\leq t\leq 1
    \end{cases}
    [/itex]
    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 [itex]\left\|\cdot\right\|_{\infty}\leq c\left\|\cdot\right\|_{s}[/itex], and of course this is true for [itex][a,b][/itex] 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.
     
  2. jcsd
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