# Tag Archives: Compactness

## Aubin-Lions Lemma

In this article, we prove the celebrated compactness lemma which will be used to show the global existence of weak solution of Navier-Stokes equation. Lemma 1. Let and be three Banach spaces with     Suppose is reflexive. Then for each , there is a constant such that     Proof. If the statement were… Read More »

## Cauchy Regular and continuity

Definition. Let be metric spaces and a function. We say is Cauchy regular provided that any Cauchy sequence in , is a Cauchy sequence in . Note that if is uniformly continuous on , then it is Cauchy regular. Indeed, let be a Cauchy sequence in and let be given. Since is uniformly continuous, there… Read More »

## Tychonoff theorem implies Axiom of choice

이번 글에서는 Tychonoff theorem이 Axiom of choice를 함의한다는 것을 보이고자 한다. Tychonoff theorem은 다음과 같다. Theorem. If is a family of compact spaces, then its product space is compact space.  이 명제의 Proof은 Axiom of choice에 의존한다. 이 글에서는 Tychonoff theorem을 증명하는 것이 목표가 아니다. 이 정리를 증명하는 방법은 다양한 형태의 Axiom of choice를 사용해서 증명하는… Read More »