Continuity of the identity function on function spaces.

[ELESSAR TELKONT]

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!

 

[foxjwill]

Try $$f_n(x)=\exp{(x/n)}$$.

Btw, I'm sure you just forgot to put this in, but the p-norm is

$$\|f\|_p = \left(\int_a^b |f|^p\,dx\right)^{1/p}.$$​