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Hello,
I need help with the following questions:
Let B = {f in C([0,1], R) | f' is continuous on (0,1), f(0)=0, and |f'(x)| <= 1}. Show that cl(B) is compact.
I'm stuck on this problem. Appreciate anyone's help on this. Thank you in advance. Thanks :)
Just remember that for aQuote:
Originally Posted by anlys
function they are equivalent:
1)is Lipschitz cont. with constant
2)is bounded by
Use this to prove thatworks in the def. of (uniform) equicontinuity of
. Now use Arzela-Ascoli.
Mean value theorem shows that B is equicontinuous .
Moreover, since, again mean value shows that, for
for each
. This yields that B is pointwise bounded (for x=1 is an easy consequence). Now Arzela Ascoli tells us that B is relatively compact, that is equivalent to the desired asertion.
By the way, the part of equicontinuity that I only mention above is better explained in Jose27 post
Thank you, Jose27 and Enrique2 for the input. I have tried using the hints that you gave me and now I can prove that the set B is equicontinuous and pointwise bounded. Thanks again :)Quote:
Originally Posted by Enrique2Mean value theorem shows that B is equicontinuous .
Moreover, since
By the way, the part of equicontinuity that I only mention above is better explained in Jose27 post
Please excuse me, it is my fault, but you have to check uniform boundedness instad of pintwiese boundedness. Anyway the given argument shows that B is in the unit ball of C([0,1])Quote:
Originally Posted by anlys