comparison between spaces

Question

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!

Answer 1

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 $L^2$?