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ELESSAR TELKONT
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Homework Statement
Show that if p\in (1,\infty) the identity functions id:C^{0}_{1}[a,b]\longrightarrow C^{0}_{p}[a,b] and id:C^{0}_{p}[a,b]\longrightarrow C^{0}_{\infty}[a,b] are not continuous.
Homework Equations
C^{0}_{p}[a,b] is the space of continuous functions on the [a,b] with the p-norm \left\vert\left\vert f\right\vert\right\vert_{p}=\int_{a}^{b}\vert f\vert^{p}\,dx
The Attempt at a Solution
It is sufficient to prove that for C^{0}[0,1] because I can map the interval [0,1] to the interval [a,b] via x=(b-a)t+a. In fact, since C^{0}[0,1] is a vector space is sufficient to prove that for f_{0}\equiv 0 the identity is discontinuous.
To prove discontinuity I have to prove that \exists \epsilon>0 \mid \forall\delta>0 I can show that \left\vert\left\vert f\right\vert\right\vert_{1}<\delta \longrightarrow \left\vert\left\vert f\right\vert\right\vert_{p}>\epsilon 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 p-norm and the \infty-norm (or the 1-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 id:C^{0}_{1}[0,1]\longrightarrow C^{0}_{\infty}[0,1] is not continuous. I have done it via the functions g_{\delta}(x)=1-\frac{1}{\delta}x for 0\leq x\leq \delta and 0 for \delta\leq x\leq 1 and it's straightforward. However this functions don't help me in my present problem. Please Help!
