# Monthly Archives: June 2015

## Equivalent condition for bounded linear operator and continuity

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오늘 실해석학 시험을 쳤는데, 다른 건 답변을 했지만(다른 것도 마이너한 실수들이 있긴 하다) 이 한 문제만 답변을 제대로 못했다.
쉬운건 알고 있었는데, 증명이 생각이 나질 않았다. 바로 증명을 보고 내 큰 실수를 또 발견했는데, ‘정의’에 충실하지 않았다는 것이다.
언제나 블로그에는 내 부끄러움도 올려야 한다고 생각하기 때문에, 못 풀었던 문제의 증명도 실어 놓는다.

Theorem. Let be a linear operator from to . Then the following are equivalent:

1. is continuous on
2. is continuous at
3. is bounded

Proof. (1) implies (2): clear.
(2) implies (3): Assume is continuous at . Then there is such that if , then we have

Then for all nonzero , we have

Since was arbitrary chosen nonzero vector, is bounded.
(3) implies (1): Assume is bounded. Choose in . Then we have

So this is Lipschitz continuous on . So it is continuous on

## Calculation of Integral with log term and some lessons

Problem. Evaluate

이 값을 구하는 과정에서 다음과 같은 결과가 필요하다.

Proposition. We have

Proof. Define where and . Consider the following contour:

Here . Then inside the contour, has simple pole at . So

So by Residue theorem, we get

First, note that

Also, we have

Note that as and as . Hence, as and , we have

Since , we get .

이제 본격적으로 문제의 적분을 계산해보도록 하자.

Solution to Problem. Define where and . Consider the following contour:

Here . Then inside the contour, has simple pole at . So

Hence, by Residue theorem, we get

(1)

Note that

(2)

Also, we have

So the essential part of our calculation is calculating

since .
Similarly,

So as and , from (1) and (2) and by Proposition, we get

Since , we get

이 문제가 오늘 시험문제에 나왔었는데, 나는 간단한 극한 계산 마무리를 그때 당시에 잘못 했었다. Estimate를 정교하지 못하게 했나라는 생각이 들었었는데, 쓸데없는 생각이 나를 막았던 것 같다.

## Application of Urysohn’s metrization theorem

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Theorem. Let be a compact metric space, a Hausdorff space, and a continuous function from onto . Then is metrizable.

Proof. Note that since and is continuous and is compact, is compact. Since is Hausdorff space, is . So is . It suffices to show is second
countable. Since is compact, it is second countable. Let be a countable basis for . Define

Clearly, is countable. For , define . Since is open, is closed. Since is closed map, is open. So if we collect these , say , we claim that is basis for .

Let be an open set in and let . Then is open and is closed. So is closed since is continuous. Since is compact space, is compact. So there is a finite collection of such that

since is a basis for and is open in . Let . Then by definition and . Moreover, . For any point , . So . This shows . Since , then . This implies for each open set in and , there is a member of such that

The second condition of basis easily comes from . This shows is a countable basis for So by Urysohn’s metrization theorem, is metrizable.