ガロア第一論文と乗数イデアル他関連資料スレ18

ガロア第一論文と乗数イデアル他関連資料スレ18 (442ﾚｽ)
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165: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/06/12(木) 22:33:50.88 ID:EWvjXceg

>>155
>2)は、必要条件を求める問題、もちろん有界閉区間での知見を「陽」に使ってよい
>っていうか「陽」につかわないって馬鹿？

ふっふ、ほっほ
下記のAI による概要で
”Theorem:
Let X and Y be metric spaces, S a subset of X, and f: S -> Y.
If f is uniformly continuous and Y is complete, then there exists a unique continuous extension of f to ¯S (the closure of S).
Furthermore, this extension is uniformly continuous.”と言ってますよ
”有界閉区間”の条件はありません！！ｗ ；ｐ）
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数学 距離空間 稠密 関数 一様連続 拡張
　↓英訳
Mathematics Metric space Dense Function Uniform continuity Extension
　↓検索 googleさん
AI による概要（AI responses may include mistakes. Learn more）
In the context of metric spaces, if a function f is uniformly continuous on a dense subset S of a complete metric space X, then f can be extended to a uniformly continuous function F defined on the entire space X. This theorem is a powerful tool for extending functions from dense subsets to the whole space while preserving uniform continuity, which is crucial in many mathematical applications.

Key Concepts and Definitions:
Metric Space:
A set equipped with a distance function (or metric) that satisfies certain properties.
Dense Subset:
A subset where every point in the larger space is either in the subset or can be approached arbitrarily closely by a point in the subset.
Uniformly Continuous Function:
A function where the distance between the function values of two points can be made arbitrarily small as long as the distance between the two points is small, regardless of where those points are in the domain.

Complete Metric Space:
A metric space where every Cauchy sequence (a sequence that gets arbitrarily close to each other) converges to a point in the space.
Theorem:
Let X and Y be metric spaces, S a subset of X, and f: S -> Y.
If f is uniformly continuous and Y is complete, then there exists a unique continuous extension of f to ¯S (the closure of S).
Furthermore, this extension is uniformly continuous.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1748354585/165

166: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/06/12(木) 22:34:28.84 ID:EWvjXceg

つづき

Proof Outline:
1. Definition of Extension:
The extension F is defined on X by considering a sequence {x_n} in S that converges to x in X. Then F(x) is defined as the limit of f(x_n) as n approaches infinity.
2. Well-Definedness:
The definition of F is shown to be independent of the chosen sequence {x_n} converging to x.
3. Continuity of Extension:
The extension F is shown to be continuous on the closure of S (i.e., ¯S).
4. Uniform Continuity of Extension:
The uniform continuity of F is established using the uniform continuity of f and the completeness of Y.

Significance:
This theorem is fundamental in analysis because it allows us to extend properties of functions defined on dense subsets to the entire space. This is particularly useful when we want to analyze the behavior of functions on a larger space using information available on a smaller, dense subset

(参考リンク：URL略す)
Extending a function by continuity from a dense subset of a space
2011/10/29 — Now, the main theorem. Theorem. Let X and Y be metric spaces, S a subset of X, and f:S→Y. If f is uniformly continuous a...
Mathematics Stack Exchange

Continuous extensions of continuous functions on dense subspaces
2012/07/12 — 1 Answer. ... Uniform continuity ensures that the Cauchy sequence (qn) in Q is mapped to a Cauchy (and hence convergent)
Mathematics Stack Exchange

Uniform continuity - Wikipedia
Continuity of a function for metric spaces and at every point of an interval (i.e., continuity of on the interval ) is expressed b...
Wikipedia, the free encyclopedia
(引用終り)
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http://rio2016.5ch.net/test/read.cgi/math/1748354585/166

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