EDB — 1K4

[1K4]Prerequisites:[1HR]↺↻,[0V3]↺↻,[0VR]↺↻,[1K2]↺↻,[1K0]↺↻.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.

1. \({\mathcal F}\) is compact

2. \({\mathcal F}\) is closed, it is equicontinuous and bounded (i.e. \(\sup _{f∈{\mathcal F}} \| f\| _∞{\lt}∞\)).