Hey - my hunch tells me that given an arbitrary open ball of functions (such as an arbitrary open ball centered at some f*) in the space of continuous functions defined on the compact interval [a,b] (ie, C0[a,b] sorry I don't know latex very well) with the standard sup norm, that this set is NOT an equicontinuous set of functions. Is this hunch true?? Anyone know how to prove this???
Let B be the open ball of radius r around f
Equicontinuity implies for all g in the open ball, and for all x in [a,b] there exists an d s.t.
lx-yl < d --> lg(x)-g(y)l < r/2
(here I set e = r/4)
but, I think it very possible to make a function h where h is in the open ball of radius r around f, but does not satisfy the above condition.
All you would have to do is ensure that on some interval of size d, you have h(x) = f(x)+r/2 and h(y) = f(y)-r/2, where f(x) >= f(y), cause then
lg(x)-g(y)l = lh(x)-h(y)+r l = h(x)-h(y)+r (since both terms are +)
h(x)-h(y)+r>= r > r/2