오늘 강의노트 보면서 정리하는데 깔끔해 보여서 여따가도 올려봄

정의는 다 빼고 적었음. 오타나 오류있으면 알려주셈


Fact. 

1. In a metric space, a subset is compact if and only if complete and totally bounded(precompact)

2. In a complete space, a subset is complete if and only if closed

3. When S is a metric space, Y is Banach space(complete), then

C_b(S,Y), the set of continuous and bounded functions from S to Y, is Banach space with sup-norm


Arzela-Ascoli theorem characterizes precompact/compact subests 

Setting. Let S be a compact metric space, and Let A ⊂ C(S,ℝ^m)


Thm. A is precompact if and only if A is pointwise bounded, equicontinuous

Lem. The followings are equivanlent.

1. A is pointwise bounded, equicontinuous

2. A is pointwise bounded, uniformly equicontinuous

3. A is bounded in C(S,ℝ^m), uniformly equicontinuous

Cor. A is compact if and only if A is equicontinuous, closed, and bounded (Fact 1)


*Note that C(S,ℝ^m) is Banach space(Fact 3), so A is complete if and only if closed(Fact2)

So, precompact sets in C(S,ℝ^m) are relatively compact (i.e. having compact closure)


*Usually, Arzela-Ascoli theorem is stated with sequentially compact conditions (convergent subsequence)


*It is used in Complex analysis in proving Riemann mapping theorem (Montel's theorem)