I was wondering. So a big part of Calculus of Variations is deciding when a functional attains a max/min. Though, I have never tried to actually do it if we are supplied with a family of functions which satisfy certain conditions couldn't we maybe use the Ascoli-Arzela theorem to prove that it is compact. Thus, for well-behaved functionals like most of the form which are continuous wouldn't the compactness of guarantee a max or min attained on ? I'm guessing that it's either A) virtually impossible to prove that a family of such functions are compact or B) it almost never happens with the norm.
Anyone have any enlightening comments? It just amazes me that I've never seen it mentioned in my book.