Let the linear operator $L:C^{\infty}_0([-1,1]) \rightarrow C(\{0\})$ be defined as $f(0)$ (evaluating a function in $C^{\infty}_0([-1,1])$ at $0.$
I would like to show that extending this to the space $L^p(-1,1)$ makes it unbounded and that $C^{\infty}_0([-1,1])$ is dense in this space.
I think I have found a simple example of a function in $L^p(-1,1)$ that shows that the linear operator is unbounded.
If we let $f(x) = M$ for $x=0$ and $f(x) = 0$ otherwise, then $f$ has $0$ $L_p$ norm, but $Tf=M$, and so there is no such $\alpha$ such that $|Tf| \le \alpha |f|_p$ for all $f \in L^p(-1,1)$. In particular, this would show that $L$ is not linear right?
The only confusion that I have is that, in this case, $f$ is in the equivalence class of the $0$ function, so it feels like a bit of a fuzzy example. But on the other hand, any member of the equivalence class should be valid as it's representative - so perhaps there are no problems here? I guess what I would like to know, is whether my example is valid?
Secondly, I am having difficulty showing that $C^{\infty}_0([-1,1])$ is dense in $L^p(-1,1)$. I know I need to show that for any $f \in L^p(-1,1)$ and any $\epsilon > 0$, I need to be able to find a $u \in C^{\infty}_0([-1,1])$ such that $||f-u||_p < \epsilon$, but I'm not really sure where to start.
I have looked at the Weierstrass approximation theorem, as well as density of other subspaces of $L^p(-1,1)$, but in each case, I seem to run into the same problem of having to prove density of some other space. A good push in the right direction, would be very helpful!
The members of $L^p[-1,1]$ are equivalence classes of functions, where $f\equiv g$ $\iff f(x)=g(x)$ except on a set of measure $0.$ We can remedy the def'n by saying that if $f\in L^p[-1,1]$ and if $f\equiv g\in C^{\infty}_0,$ then $L(f)=g(0)$, as there can be only one such $g.$
If $1\leq p<\infty$ then the operator $L$ is already unbounded on $S_p=\{[f]_p:f\in C^{\infty}_0[-1,1]\}$, where $[f]_p=\{g\in L^p[-1,1]:g\equiv f\}$, because for any $n\in N$ there exists $f_n\in C^{\infty}_0[-1,1]$ with $f_n(0)=1$ and $\int_{-1}^1 |f(x)|\;dx<1/n^p.$