Dear professors,
I have some trouble with an exercise in a functional-analysis book.
The exercise seems to be about the Arzela-Ascoli theorem.
Let H^s(D) be the s-Sobolev space with s > n/2 and
S is a subset of H^s(D) such that
sup_{f in S} |f|_{s} =< M < +oo, (1)
where | |_{s} is defined by
|f|_{s}^{2} = int_{D} |f(x)|^{2} (1 + |x|^{2})^(s) dx.
(D is a subset of R^n)
The problem seems to make me derive the uniform boundedness of
some set of bounded continuous functions under the norm
| |_{oo} from (1) above.
I have thought about this for four days, but without success.
Of course, there seem to be many examples such that both of
lim_{f in S} |f|_{oo} =< M
and (1) are satisfied. And so, I have been very confused!
Any hint/help would be extremely welcome.
Benjamin
I have some trouble with an exercise in a functional-analysis book.
The exercise seems to be about the Arzela-Ascoli theorem.
Let H^s(D) be the s-Sobolev space with s > n/2 and
S is a subset of H^s(D) such that
sup_{f in S} |f|_{s} =< M < +oo, (1)
where | |_{s} is defined by
|f|_{s}^{2} = int_{D} |f(x)|^{2} (1 + |x|^{2})^(s) dx.
(D is a subset of R^n)
The problem seems to make me derive the uniform boundedness of
some set of bounded continuous functions under the norm
| |_{oo} from (1) above.
I have thought about this for four days, but without success.
Of course, there seem to be many examples such that both of
lim_{f in S} |f|_{oo} =< M
and (1) are satisfied. And so, I have been very confused!
Any hint/help would be extremely welcome.
Benjamin