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Continuity of the identity function on function spaces.

  1. Sep 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that if [tex]p\in (1,\infty)[/tex] the identity functions [tex]id:C^{0}_{1}[a,b]\longrightarrow C^{0}_{p}[a,b][/tex] and [tex]id:C^{0}_{p}[a,b]\longrightarrow C^{0}_{\infty}[a,b][/tex] are not continuous.

    2. Relevant equations

    [tex]C^{0}_{p}[a,b][/tex] is the space of continuous functions on the [a,b] with the p-norm [tex]\left\vert\left\vert f\right\vert\right\vert_{p}=\int_{a}^{b}\vert f\vert^{p}\,dx[/tex]

    3. The attempt at a solution

    It is sufficient to prove that for [tex]C^{0}[0,1][/tex] because I can map the interval [tex][0,1][/tex] to the interval [tex][a,b][/tex] via [tex]x=(b-a)t+a[/tex]. In fact, since [tex]C^{0}[0,1][/tex] is a vector space is sufficient to prove that for [tex]f_{0}\equiv 0[/tex] the identity is discontinuous.

    To prove discontinuity I have to prove that [tex]\exists \epsilon>0 \mid \forall\delta>0[/tex] I can show that [tex]\left\vert\left\vert f\right\vert\right\vert_{1}<\delta \longrightarrow \left\vert\left\vert f\right\vert\right\vert_{p}>\epsilon[/tex] or what's the same, there is a sequence of functions that the area below their absolute value is less than delta but the area below their absolute value elevated to p is not bounded by epsilon. For the second case I must prove that there exists a sequence of functions for that the maximum of each is not bounded by epsilon but the area below their absolute value elevated to p is bounded by delta.

    In other words I have to prove that the [tex]p[/tex]-norm and the [tex]\infty[/tex]-norm (or the [tex]1[/tex]-norm) are not equivalent.

    My problem is that to prove vía sequences I can't figure out a function sequence that elevated to p can help me to prove that.

    I have already proven something similar: that [tex]id:C^{0}_{1}[0,1]\longrightarrow C^{0}_{\infty}[0,1][/tex] is not continuous. I have done it via the functions [tex]g_{\delta}(x)=1-\frac{1}{\delta}x[/tex] for [tex]0\leq x\leq \delta[/tex] and 0 for [tex]\delta\leq x\leq 1[/tex] and it's straightforward. However this functions don't help me in my present problem. Please Help!
     
  2. jcsd
  3. Sep 6, 2009 #2
    Try [tex]f_n(x)=\exp{(x/n)}[/tex].

    Btw, I'm sure you just forgot to put this in, but the p-norm is
    [tex]\|f\|_p = \left(\int_a^b |f|^p\,dx\right)^{1/p}.[/tex]​
     
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