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2021/09/05  阅读：32  主题：默认主题

# Amann Analysis Chapter I Notes

### 前言：一些碎碎念

1）不够 self-contained：例如，baby Rudin 一上来就在讨论 的无理性。然而，此时连有理数是什么都没有定义。如果把有理数定义为两个整数之比，那么又产生了新的问题：什么是整数？虽然这些问题相对而言并不是分析学所考察的范畴，但缺少了他们，会使得读者感到缺少了基础。我认为，数学分析作为大多数人第一次接触到严谨的数学的机会，应该是 axiomatic 的，即规定一定的公理和不加定义的概念，其他一切结果均只由前面的结果推出。

2）语言晦涩、生硬：一众国外课本的翻译版都是如此，我曾部分看过的 Zorich 的数学分析和菲赫金哥尔茨的《微积分学教程》尤甚。这些译者仿佛是从两百年前的古代穿越回来的，大量运用晦涩的、文言文式的表达。我看它们的顺畅程度甚至不如看英文版的。此外，诸如 baby Rudin 这样的经典教程，其往往缺少了必要的“废话”，例如定义的动机、定理的解释和应用等等，而只是 axiom-definition-lemma-proposition-theorem-corollary 的枯燥结构。这很容易导致读者失去兴趣，因为他们看不到这些定义与定理的用途。

3）不够 general：一般的数学分析教程只研究最基本的 ，而不讲述其他更加一般的代数结构、空间等。这导致与后续课程的衔接不够顺畅，例如实分析、泛函分析等。一般来说，数学分析是和高等代数/抽象代数并行教学的，故也应当使用一些代数学的基本工具。如果从一开始就讲述某个结论最一般的形式，即使更加抽象，其优势也是显而易见的。（引用一句话："Abstractness is the price of generality."）

Amann 的数学分析三卷本在网上鲜有提及，但它是我认为的最好的分析学教程。它是布尔巴基式的一套分析课本：第一册的第一章（Foundations）假设读者毫无基础，从最基本的数理逻辑与集合论讲起，用皮亚诺公理定义自然数，并逐步延拓到最一般的数系——复数，期间引入了各种代数结构，如群环域。随后的几章介绍了收敛、连续、一元函数的微分和函数列等基本概念。第二册详细介绍了一元和多元的微积分，且并不局限于实数域：在必要的时候，它会将结果拓展到复数域甚至更一般的代数结构。第三册则介绍了测度论、流形和其上的微积分等实变函数内的基本概念，并最终以一般化的斯托克斯公式结尾。

### 2 Sets：集合

ZFC 中的 ZF 指的是 Zermelo–Fraenkel，两位发明者；C 指的是 axiom of choice，即选择公理。ZF 由八条公理组成，C 因为有一定争议（会引起诸如 Banach-Tarski paradox 之类的违反直觉的结论），有时并不被承认。不过，C 与佐恩引理 (Zorn's lemma)、良序原理 (well-ordering principle) 等有用的结论等价。

1. The Axiom of Extensionality

This axiom establishes the most basic property of sets: a set is completely characterized by its elements alone. Statement: Two sets and are equal if and only if the statements ( is an element of ) and ( is an element of ) are equivalent.

1. The Empty Set Axiom

This axiom ensures that there is at least one set. Statement: There exists a set (called the empty set and denoted ) which contains no elements.

1. The Axiom of Subset Selection

This axiom declares subsets of a given set as sets themselves. Statement: Given a set , and a formula with one free variable, there exists a set whose elements are precisely those elements of which satisfy .

1. The Power Set Axiom

This axiom allows us to construct a bigger set from a given set. Statement: For every set , there exists a set, called the power set of (denoted ), containing exactly the subsets of .

1. The Axiom of Replacement

This axiom allows us, given a set, to construct other sets of the same size. Statement: Given a set and a functional predicate in the language of set theory, there is a set which consists of exactly those elements related to elements in .

1. The Axiom of Union

This axiom allows us to take unions of two or more sets. Statement: Given a set , there exists a set with exactly those elements which belong to some element of .

1. The Axiom of Infinity

This gives us at least one infinite set. Statement: There exists an infinite set, i.e., a set and an injection which is not bijective.

1. The Axiom of Foundation

This makes sure no set contains itself, thus avoiding certain paradoxical situations. Statement: The relation belongs to is well-founded. In other words, for every nonempty set , there exists a set which is disjoint from .

1. The Axiom of Choice

This allows to find a choice set for any arbitrary collection of sets. Statement: For each collection of disjoint sets, there exists a set (called the choice set) containing precisely one element of each set in the collection.

### 7. Groups and Homomorphisms; 8. Rings, Fields and Polynomials：群环域 etc.

Group 是带有一个 operation 的集合，where is associative and has both an identity and an inverse.

Ring 是一个 Abelian group with an associative operation (multiplication), 符合 distributive law.

Field 是一个 commutative ring with unity，且其上的两个运算具有不同的 inverse, 且去掉了 后的集合加上 multiplication 也构成 Abelian group.

### 12. Vector Spaces, Affine Spaces and Algebras：向量空间、仿射空间与代数

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2021/09/05  阅读：32  主题：默认主题