Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

There a lot of function spaces and would be nice if somebody can correct me if I am wrong in comparing a few. I want to compare $C^2,L^2,W^{2,2}$ (continuous up to third derivative, Hilbert space of square integrable functions and Sobolev with second weak derivative in $L^2$) of functions defined on a compact set. I think $C^2\subset W^{2,2}\subset L^2$. Is that the right inclusion? thanks!

share|improve this question
    
In the case of open bounded sets of $\mathbb{R}^n$, yes, this is true. –  matgaio Apr 25 '12 at 20:09
    
I guess you cold take a look at the Evans's "Partial Differential Equations", chapter 5. There you will find lots of inclusions of space of functions as linear bounded (and sometimes compact) operators between these spaces. –  matgaio Apr 25 '12 at 20:11

1 Answer 1

up vote 1 down vote accepted

Yes, this is correct. Note that a continuous function on a compact set is square integrable (because it is bounded and a compact set has finite measure), and if the classical derivative exists then the weak derivative exists and they are equal. Thus $C^2$-functions are twice weakly differentiable and the derivatives are again square integrable because they are continuous. In other words, $C^2 \subset W^{2,2}$. Functions in $W^{2,2}$ are square integrable by definition, so we have $W^{2,2} \subset L^2$.

An interesting question for you to ponder is: are the inclusion maps continuous? That is: is the map $f \mapsto f$ bounded from $C^2$ to $W^{2,2}$ and from $W^{2,2}$ to $L^2$?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.