18.1 On Ascoli–Arzelà’s Theorem
Now we’ll see some exercises that reconstruct the famous Ascoli–Arzelà Theorem.
- E406
[1JX]Prerequisites:5 subpoint 6,7. Let \(I⊆ ℝ\) be a subset. Let \(X\) be the set of functions \(f:I→ℝ\) bounded and uniformly continuous. We equip \(X\) with distance \(d_∞(f,g)=\| f-g\| _∞\). Show that the metric space \((X,d_∞)\) is complete. Hidden solution: [UNACCESSIBLE UUID ’1JY’] In particular, \(X\) is a closed vector subspace of the space \(C_ b(I)\) of continuous and bounded functions.
Define \((X,d_∞)\) as in the previous exercise 1. Fix now \({\mathcal F}⊆ X\) a family of functions, suppose \({\mathcal F}\) is totally bounded (as defined in 299): Show then that the family \({\mathcal F}\) is equicontinuous.
Hidden solution: [UNACCESSIBLE UUID ’1K1’]
[1K2]Prerequisites:405,5, 6.3.Difficulty:*.Let now \(I⊆ ℝ\) be a closed and bounded interval. Let \(f_ n:I→ℝ\) continuous functions, and suppose that the sequence \((f_ n)\) is equicontinuous and bounded (i.e. \(\sup _ n \| f_ n\| _∞{\lt}∞\)). Show that there is a subsequence \(f_{n_ k}\) that converges uniformly.
Hidden solution: [UNACCESSIBLE UUID ’1K3’]
[1K4]Prerequisites:405,299,9,1,1.Difficulty:**.Note:A version of Ascoli–Arzelà’s theorem.Let \(I⊆ ℝ\) be a closed and bounded interval. Let \(C(I)\) be the set of continuous functions \(f:I→ℝ\). We equip \(C(I)\) with distance \(d_∞(f,g)=\| f-g\| _∞\). We know that metric space \((C(I),d_∞)\) is complete.
Let \({\mathcal F}⊆ C(I)\): the following are equivalent.
\({\mathcal F}\) is compact
\({\mathcal F}\) is closed, it is equicontinuous and bounded (i.e. \(\sup _{f∈{\mathcal F}} \| f\| _∞{\lt}∞\)).