# Baby Rudin Chapter 1-6 总结

May 23, 2019

## Chapter 1 The Real and Complex Number Systems

### Selected Definitions and Theorems

• 1.7 Definition $E$ is bounded above if there is $y \in S$ s.t $x \leq y$ for all $x \in E$. We say that $y \in S$ is an upper bound for $E$ if $x \leq y$ for all $S \in E$.
• 1.8 Definition Suppose S is an ordered set, $E \subset S$, and $E$ is bounded above. Suppose there exists an $\alpha \in S$ and $\alpha$ is the least upper bound if for each upper bound $y$ for $E$ we have $\alpha \leq y$ and $\alpha$ is an upper bound for $E$.
• 1.10 Definition A linear ordered set $S$ has the least upper bound property if for each $E \subset S$ s.t. $E$ is bounded above and is non empty, then $E$ has a least upper bound.
• 1.11 Theorem $S$ an ordered set with the least upper bound property. Let $B \subset S$, $B \neq \emptyset$ and bounded below. Let $L=$ all lower bounds of $B$. Then $L$ is bounded above, non empty and so $\alpha = \sup L$ exists and $\alpha$ is the greatest lower bound for $B$.
• 1.20 Theorem

• Archimedean Property If $x, y \in \mathbb{R}$, $x > 0$, then there exists a natural n s.t $nx > y$
• $\mathbb{Q}$ is dense in $\mathbb{R}$ if $x, y \in \mathbb{R}$, $x < y$, then there exists $p \in \mathbb{Q}$ s.t. $x < p < y$
• 1.37 Theorem (Schwarz Inequality) Suppose $x, y, z \in \mathbb{R}^k$, then $|x \cdot y| < |x|\cdot|y|$

## Chapter 2 Basic Topology

### Summary

2.30-2.37围绕紧致性 (Compactness)这一概念进行讨论。从这里开始本章的内容开始变得抽象了起来，个人理解这是出于紧致性本就是一个杜撰出来的概念，有点类似平面几何中的辅助线。但这个概念非常重要，他的作用相当于把有限和无限、开和闭联系了起来，通过引入这一概念能够得到一些非常好的性质，例如在度量空间 $X$ 上紧致的子集一定在 $X$ 上是闭集和一个有界的闭集一定紧致等等，在之后章节讨论映射和连续性的时候也可以反映这一点。

2.38-2.42 将集合与区间对应起来，并把本章之前的内容从 $\mathbb{R}$ 推广到了 $\mathbb{R}^k$ 上。最后关于 Perfect Set 的内容 MATH 4130 没有涉及，故不再赘述。

### Selected Definitions and Theorems

• 2.4 Definition $J_n = {1,\ldots,n}$, $n \in \mathbb{N}$

• $A$ finite if $A \sim J_n$ for some $n$
• $A$ infinite if $A$ is not finite
• $A$ countable if $A \sim \mathbb{N}$
• $A$ at most countable if $A$ countable or finite
• $A$ uncountable if $A$ is not at most countable
• 2.15 Definition A metric space $(x, d)$ is a set $X$ and a function $d: X\times X \rightarrow \mathbb{R}$ s.t.

• $d(p, q) > 0$ if $p,q \in X$, $p \neq q$; $d(p, p) = 0$
• $d(p,q) = d(q, p)$
• $d(p, q) < d(p, r) + d(r, q)$, for any $r \in X$
• 2.18 Definition Let $X$ be a metric space

• A neighborhood of a point $p$ is a set $Nr(p)$ consisting of all points $q$ such that $d(p, q) < r$. The number $r$ is called the radius of $Nr(p)$
• A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q \neq p$ such that $q \in E$
• $E$ is closed if every limit point of $E$ is a point of $E$
• A point $p$ is an interior point of $E$ if there is a neighborhood $N$ of $p$ such that $N \subset E$
• $E$ is open if every point of $E$ is an interior point of $E$
• 2.32 Definition A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover.

## Chapter 3 Numerical Sequences and Series

### Selected Definitions and Theorems

• 3.2 Theorem Let $(X, d)$ be metric space, $E \subseteq X$, $p \in X$, $p$ is a limit point of $X$ iff there is a sequence $p_n \in E$ s.t. $\lim_{n \to \infty} p_n \rightarrow p$ and $p_n \neq p$ for all $n$.
• 3.8 Definition A sequence {$p_n$} in a metric space $(X, d)$ is said to be a Cauchy sequence if for every $\epsilon > 0$ there is an integer $N$ such that $d(p_n , p_m) < \epsilon$ if $n > N$ and $m > N$.
• 3.9 Definition For $E \subseteq X$, $(X, d)$ a metric space, {% raw %}$Diam(E) = \sup \{d(x, y)|x, y \in E\}${% endraw %}.

## Chapter 4 Continuity

### Summary

4.5之后的内容则是本章的重点——连续性了。其实连续性定义的核心也是 $\delta - \epsilon$ 法则，一系列相关的性质都可以从这一点着手进行证明，比如连续函数的加减乘除和复合都是连续的。从4.5到4.12都是这些基础性质的讨论。4.13开始将连续性和紧致性联系起来，是本章的点睛之笔。其实在之前的定理4.8就已经介绍了从集合的角度如何定义连续性，从而无需涉及极限，使许多集合上诸如紧致性等一些很好的性质得以利用起来。4.13之后的一系列定理则正是这一点的具体示例，例如4.14说明定义域是紧致集的连续函数的值域也是紧致集; 4.18定义了一致连续，介绍了一个更加苛刻的连续性定义；4.19通过紧致性把连续和一致连续建立了联系，定义域是紧致集的连续函数一致连续。这一部分的内容非常精彩，应当反复研读并认真完成对应习题。

### Selected Definitions and Theorems

• 4.8 Theorem For $f: X \rightarrow Y$, $f$ is continuous iff $f^{-1}(V)$ is open for each $V \subseteq Y$ open.
• 4.14 Theorem For $f: X \rightarrow Y$ continuous, $X$ compact then $f(X)$ is compact.
• 4.18 Definition For $f: X \rightarrow Y$, we say that $f$ is uniformly continuous on $X$ if for every $\epsilon > 0$ there exists $\delta > 0$ s.t. $d_Y(f(p), f(q)) < \epsilon$ for all $p$, $q$ in $X$ for which $d_X(p, q) < \delta$.
• 4.19 Theorem For $f: X \rightarrow Y$ continuous, $X$ compact then $f(X)$ is uniformly continuous.

## Chapter 5 Differentiation

### Selected Definitions and Theorems

5.19 Theorem Suppose $f: [a, b] \rightarrow \mathbb{R}^k$ continuous, $f$ differentiable on $[a, b]$, then there exists $x \in [a, b]$ s.t. $|f(b) - f(a)| < (b - a) |f'(x)|$ .

## Chapter 6 The Riemann-Stieltjes Integral

### Summary

6.8-6.11主要对可积性的判断进行讨论，6.12-6.19则都是关于黎曼积分性质的各种定理，但从6.8-6.19基本都是运用了黎曼积分的定义，尤其是6.4和6.6，进行证明。把这些证明弄懂对摸清本章习题的解题套路很有帮助，这些定理本身的内容也一定都要熟记。通过这一部分我们也可以发现函数的连续性和单调性(递增)对函数是否可积与黎曼积分的一些性质是否成立都相当重要，并且有些时候被积函数和积分变量可以相互转换。

### Selected Definitions and Theorems

• 6.4 Theorem {% raw %}If $P^*$ is a refinement of $P$, then $L(P, f, a) \leq L(P^*, f, a)$, $U(P, f, a) \geq U(P^*, f, a)$. {% endraw %}
• 6.6 Theorem $f \in R(\alpha)$ on $[a, b]$ iff $\forall \epsilon > 0$ $\exists P$ a partition of $[a, b]$ s.t. $U(P,f,a) - L(P,f,a) < \epsilon$.
• 6.8 Theorem If $f$ is continuous on $[a, b]$ then $f \in R(\alpha)$ on $[a, b]$.
• 6.17 Theorem $\alpha: [a, b] \rightarrow \mathbb{R}$ monotonically increasing and differentiable, $\alpha' \in R$ on $[a, b]$, $f$ bounded on $[a,b]$. Then $f \in R(\alpha)$ iff $f\alpha' \in R$ and $\int_{a}^{b}fd\alpha = \int_{a}^{b} f(x)\alpha'(x)dx$.
• 6.21 The fundamental theorem of calculus If $f \in R$ on $[a, b]$ and if there is a differentiable function $F$ on $[a,b]$ s.t. $F' = f$, then $\int_{a}^{b}f(x)dx = F(b) - F(a)$.

note.pdf

Ch1:

Ch2:

Ch4: