Terence Tao Analysis

This repo contains notes of the following textbooks:

Analysis I 3rd edition by Terence Tao.
Analysis II 3rd edition by Terence Tao.

The notes cover almost all axioms, definition, corollaries, examples, lemmas, propositions, remarks, theories, meta-proofs, proofs, notes and exercises appear in the textbooks, which are copied directly from the books, with or without some modifications.

The notes also have some additional corollaries to help proving others.

Contribution

Issues

I'm not a native English user, and do not have mathematical background. So any helps are appreciated, especially for the following things:

Syntax and grammer correction.
Making proofs more rigorous.
Proofs I am struggled with.

Rules

Please fork the repository and submit a pull request.

Math Symbols, Equations and Statements

All inlined math symbols and equations must be enclosed by \( and \).

Let \(x\) be a real number % enclose variable x
\(x = 1\)                  % enclose equation x = 1

If variables are seperated by comma with the same properties, then enclosed them together.

Let \(x, y\) be a real number % enclose comma seperated variables

All block-style single line equation and statement must be enclosed by \[ and \].
\[ must followed by a new line, and \[ must following a new line.

\[                            % followed by a new line
  \lim_{n \to \infty} a_n = c
\]                            % following a new line

All block-style multi-line equations and statements must be enclosed by the environment \begin{align*} and \end{align*}.

\begin{align*} % enclose multi-line equations by align
a &= 1 \\
b &= 2
\end{align*}

Theorem-like Statements

All theorem-like statements must be enclosed by the proper environment.
All theorem-like statements must be correctly labeled.

% for axiom
\begin{axiom}[axiom name]\label{chapter_no.axiom_no}
% axiom goes here
\end{axiom}

% for definition
% similarly for corollary, example, lemma, proposition, remark, theorem
\begin{definition}[definition name]\label{chapter_no.section_no.statement_no}
% definition goes here
\end{definition}

% for exercise
\begin{exercise}[exercise name]\label{ex chapter_no.section_no.exercise_no}
% exercise goes here
\end{exercise}

% for additional corollary
\begin{additional corollary}\label{ac chapter_no.section_no.corollary_no}
% additional corollary goes here
\end{additional corollary}

% for proof
% similarly for meta-proof, note
\begin{proof}
% proof goes here
\end{proof}

All references to theorem-like statements must be enclosed by \ref{}, and specify the type of theorem-like statement with capital initial.

Axiom \ref{2.1}                     % reference axiom
Definition \ref{2.2.1}              % reference definition
Corollary \ref{2.2.9}               % reference corollary
Example \ref{5.5.4}                 % reference example
Lemma \ref{2.2.2}                   % reference lemma
Proposition \ref{2.2.4}             % reference proposition
Remark \ref{2.2.15}                 % reference remark
Theorem \ref{5.5.9}                 % reference theorem
Exercise \ref{ex 2.2.1}             % reference exercise
Additional Corollary \ref{ac 2.2.1} % reference additional corollary

All statements must be enclosed either by the environment \begin{note} and \end{note} or by other theroem-like statements.
If there are some statements which might help prove others and must be included, use the environment \begin{additional corollary} and \end{additional corollary}.
- Additional Corollaries must also come with proof.

Quotes and Emphasis

Single Quotes must be enclosed by ` and '.
Double Quotes must be enclosed by `` and ''.

This is a `single quote' example.
This is a ``double quote'' example.

Emphasis text must be enclosed by \emph{}.

This is a \emph{emphasis} example.

Operators, Subscripts and Superscripts

Binary operator must be surrounded by spaces.

\(a + b\) % + is surrounded by spaces

Multiplication symbol can be ignored, and when doing so, no space allowed in between operands.

\(a \times b\) % \times is surrounded by space
\(ab\)         % no space in between

Subscript take precedence over superscript.
If there is only single letter in subscript and superscript, then do not enclose with { and }.

a_n^2                      % do not enclose with {}
\((a_n)_{n = 1}^{\infty}\) % enclosed by {}

If subscript and superscript are followed with aother symbols, then appended with additional space.

\(a_n b_n\)
\(x^2 y^2\)

Quantifiers

Quantifiers must be followed with additional space.

\(\exists\ n \geq 1\) such that ...
\(\forall\ n \geq 1\) we have ...

If there are text before Quantifiers, then prepended with additional space.

\(\exists\ \alpha \in I \ \exists\ \beta \in J\)
\(n \geq 1 \ \forall\ n \in \{1,2,3\}\)

Absolute Value, Floor, Ceiling

Absolute value must be written as \abs*{}.

\(\abs*{a_j - a_k}\)

Floor must be written as \floor*{}.

\(\floor*{x}\)

Ceiling must be written as \ceil*{}.

\(\ceil*{x}\)

Enumerate

\item in enumerate environments must be indented.

\begin{enumerate}
  \item statement 1
  \item statement 2
\end{enumerate}

Labels must be enclosed by ( and ).

\begin{enumerate}[label=(\roman*)]
  \item statement 1
  \item statement 2
\end{enumerate}

Nested enumerate environments' labels must follow the order: \alph, \Roman, \roman and \arabic.

\begin{enumerate}[label=(\alph*)]
  \item alph statement 1
  \begin{enumerate}[label=(\Roman*)]
    \item Roman statement 1
    \begin{enumerate}[label=(\roman*)]
      \item roman statement 1
      \begin{enumerate}[label=(\arabic*)]
        \item arabic statement 1
        \item arabic statement 2
      \end{enumerate}
      \item roman statement 2
    \end{enumerate}
    \item Roman statement 2
  \end{enumerate}
  \item alph statement 2
\end{enumerate}

Unsolved Exercise

Analysis I
- 8.1.1
- 8.1.9
- 8.1.10
Analysis II
- not start yet

mcps5601/terence-tao-analysis