Given the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$, I need to determine (and be able to justify) whether it is compact in the following three cases:
- $T: C^{1}[0,1]\mapsto C[0,1]$
- $T: C^{2}[0,1]\mapsto C^{1}[0,1]$
- $T: C^{2}[0,1]\mapsto C[0,1]$.
Although ideally I'd like to see full solutions on all of these - I'm trying to teach myself, and oftentimes the best way for me to do so is to dissect solutions and work backwards.
However, I suppose I could live knowing if the answer is yes or no for all three, and in detail how to do one of them, if of course, seeing how to do one of them would, in fact, help me do the other ones.
What I know about compact operators is the following: I know that they map bounded sets into compact sets (although I've also heard pre-compact sets; not sure 100% if that's true, though). I know that the image of a bounded sequence under a compact operator has a convergent subsequence.
Then, I thought perhaps I could use the Arzela-Ascoli Theorem (Let $X$ be a compact metric space and $\{f_{n}\}$ a uniformly bounded equicontinuous sequence of real-valued functions on $X$. Then $\{f_{n}\}$ has a subsequence that converges uniformly on $X$ to a continuous function $f$ on $X$) to help me, but I'm not sure how.
I've also heard a bit about weakly-convergent/strongly-convergent, although I don't think I have quite as good a handle on what that means as I should have.
On the other hand, what I have learned about the differential operator from my own reading isn't terribly helpful, because I've only ever seen it used in $L^{2}[a,b]$, not in $C^{n}[a,b]$ so I'm not sure how to apply it here.
I understand there's a lot going on in this question, but I am extremely clueless about this, need to know it, and would be so incredibly grateful if someone would please humour me and share with me this knowledge. Thank you.
For $1$, start with any bounded sequence $\{ f_n \}$ in $C[0,1]$ that has no convergent subsequence. Then $F_n\int_{0}^{x}f_n(t)dt$ gives you a bounded sequence $\{ F_n \} \subset C^1[0,1]$ whose image under $T$ is $\{ f_n \}$.
For $2$, the same technique holds. Start with a bounded sequence in $C^1[0,1]$ with no convergent subsequence, and integrate.
For $3$, a bounded sequence $\{ f_n \} \subset C^2[0,1]$ has uniformly bounded first and second derivatives, and is mapped by $T$ to a sequence $\{ f_n' \}$ of bounded functions in $C[0,1]$ with uniformly bounded derivatives, which is an equicontinuous subset of $C[0,1]$. Apply Arzela-Ascoli.