Self-study Notes: Analysis

Book(s) used: Principles of Mathematical Analysis- Rudin (Baby Rudin)

Rudin Chapter 1. Real & Complex Number Systems

In proving "For positive rational $p$ such that $p^2<2$ or $p^2>2$, there's always a closer rational $q$ to $\sqrt2$", the way he came up with the \( \frac{p^2-2}{p+2}\) as a margin is interesting. It doesn't disturb the flow of understanding but I want to know how he gets it.

Least upper bound/ supremum is denoted as sup S; greatest lower bound/ infimum is denoted as inf S (S is an ordered set).

Ordered field: the additive identity defines negativity/positivity

The real field

Existence theorem: There exists an ordered field $R$ with the least upper bound property; $R$ contains $Q$ as a subfield.

for $x\in R$, $y\in R$,
archimedean property: if $x>0$, $\exists n \in N$ such that $nx>y$
$Q$ is dense in $R$: $\exists p \in Q$ such that $x<p<y$

Proving the existence and uniqueness of positive real $y$: $y^n=x$, prove $y=\sqrt[n] x$ is the least upper bound of {$t\in R^+|t^n<x$}

Extended real number system

Is not a field
For every $x \in R$, $-\infty < x < +\infty$
$+\infty$ is an upper bound for every subset of $R$, $-\infty$ is an lower bound for every nonempty subset of $R$.

The complex field

Define a complex number as an ordered pair of real numbers. Defined addition and multiplication algebraically.

Is a field; $(0,0)$ additive identity; $(1,0)$ multiplicative identity.

Define $i=(0,1)$.

Notation for $z=a+bi$: $a=Re(z)$, $b=Im(z)$.
Conjugate of $z$: $\overline{z}$
Absolute value of $z$: $|z|=\sqrt{ z\overline{z}}$

Schwarz inequality

$|\sum\limits_{j=1}^{n} a_j \overline{b}_j|^2 \le \sum\limits_{j=1}^{n} |a_j|^2 \sum\limits_{j=1}^{n} |b_j|^2$
I also found many other forms of this inequality on the Internet, most without the conjugate. I assume those only refers to the real.

Euclidean k-space

$R^k$: set of tuples $(x_1,x_2,...,x_k)$, $x\in R$, $k\in N$.
norm $|x|=\sqrt{x\cdot x}$
$|x\cdot y|\le|x||y|$ (Schwarz inequality for reals)
$|x+y|\le|x|+|y|$
$|x-z|\le|x-y|+|y-z|$ (metric space)

[exercises] Oof, I'm a little stuck on the first ten questions. It's expected, but I may read other texts before trying to solve questions that I'm not capable of.

Rudin Chapter 2. Basic Topology

I've hear that Munkres' book is more "studyable" for the introductionary content, for Rudin is too concise. I might put most notes on the other page.