preface.tex

\chapter*{Preface}

These are my notes on the book ``Analysis I'' and ``Analysis II,'' 4th edition, written by Terence Tao.
My notes cover statements I found important or interesting, including almost all axioms, corollaries, exercises, lemmata, proofs, propositions, remarks, and theories in both books.
Errata on Tao's website (\url{https://terrytao.wordpress.com/books/analysis-i} and \url{https://terrytao.wordpress.com/books/analysis-ii}) are used to correct errors in both books.
I have written proofs for exercises.
Most exercises cannot be completed without referencing solutions from online forums.
So, many thanks to the people who their provided solutions at \url{https://math.stackexchange.com}, \url{https://taoanalysis.wordpress.com}, and many other sources.

I wrap each statement in a \LaTeX\ environment.
Statements shows up in the books already have an environment wrapped around them.
These statements are wrapped in the same environment as in the book, but with different typesets (names).
The following table provides the mapping between typesets and the meaning of each environment.
The only environment does not show up in the books is ``additional corollaries.''
It is used to wrap statements that I added to help proving other complex statements.
\begin{table}[h]
  \centering
  \begin{tabular}{|c|c|}
    \hline
    Environment Typesets  & Meaning              \\
    \hline
    \namecref{i:2.1}      & Axiom                \\
    \hline
    \namecref{i:ac:2.2.1} & Additional Corollary \\
    \hline
    \namecref{i:ch:1}     & Chapter              \\
    \hline
    \namecref{i:2.2.9}    & Corollary            \\
    \hline
    \namecref{i:2.2.1}    & Definition           \\
    \hline
    \namecref{i:3.1.10}   & Example              \\
    \hline
    \namecref{i:ex:2.2.1} & Exercise             \\
    \hline
    \namecref{i:2.2.2}    & Lemma                \\
    \hline
    Note                  & Note                 \\
    \hline
    \namecref{i:2.2.4}    & Proposition          \\
    \hline
    \namecref{i:sec:2.1}  & Section              \\
    \hline
    \namecref{i:3.6.12}   & Theorem              \\
    \hline
  \end{tabular}
  \caption{Mapping between environment typesets and meaning.}
\end{table}