UPDATE: More idiomatic variants in 1_7, 2_15 and Notations.

Author: gitcordier

FA_DM.pdf

@@ -102,7 +102,7 @@ \subsection{Justifying the terminology}
 % First proof: Use the given hint
 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
 \subsection{Proof (with the given hint)}
-We now prove the second part by constructing a specific sequence $\singleton{f_n}$ that simultaneously satisfies (a) and (b). %
+We now prove the second part by constructing a specific sequence $\singleton{f_n}$ that satisfies both (a) and (b). %
 Indeed, the hint suggests that there exists a bijective mapping % 
 %
 \begin{align}
@@ -131,7 +131,7 @@ \subsection{Proof (with the given hint)}
  \tendsto{n}{\infty} \infty, 
 \end{align}
 %
-given a sequence $\singleton{\gamma_n}$ that tends to $\infty$. This proves (b), since %
+given a sequence $\singleton{\gamma_n}$ diverging to $\infty$. This proves (b), since %
 $\singleton{\gamma_n f_n(x_\gamma)}$ diverges.
 \end{proof}
 %
@@ -150,7 +150,7 @@ \subsection{Proof with binary expansions (no hint)}
  (\beta_1, \dots, \beta_n, \dots) & \mapsto \sum_{k=1}^\infty \beta_k 2^{\minus k}. \nonumber
 \end{align}
 %
-A suitable $\singleton{f_n}$ can be defined under the same notation, as follows: %
+A suitable $\singleton{f_n}$ can be defined as follows: %
 %
 \begin{align}
  \label{definition of f_n(alpha)} f_n(x) \Def \begin{cases} 
@@ -168,14 +168,19 @@ \subsection{Proof with binary expansions (no hint)}
  n_{k+1} - n_k > k.
 \end{align}
 %
-The point is that $1_{\{n_1, n_2, \dots \}}$ is eventually aperiodic on ${\N_{+}}$. Moreover, the specialization %
-$\beta^{\gamma} = 1_{\{n_1, n_2, \dots \}}$ implies %
+The crucial point is that the sequence $1_{\{n_1, n_2, \dots \}}$ is eventually aperiodic. Moreover, the particular choice %
+%
+\begin{align}
+ \beta^{\gamma} \Def 1_{\{n_1, n_2, \dots \}}
+\end{align}
+%
+implies %
 %
 \begin{align} \label{p sum of bits}
  \beta^\gamma_1 + \dots + \beta^\gamma_{n_1} + \dots + \beta^\gamma_{n_k} = k.
 \end{align}
 %
-Finally, the combination of (\ref{definition of f_n(alpha)}) with (\ref{p sum of bits}) yields %
+Finally, (\ref{definition of f_n(alpha)}) and (\ref{p sum of bits}) together yield %
 %
 \begin{align}
  \gamma_{n_k} f_{n_{k}}(\bin(\beta^\gamma))
@@ -185,7 +190,7 @@ \subsection{Proof with binary expansions (no hint)}
  \tendsto{k}{\infty}\infty.
 \end{align}
 %
-In conclusion, every sequence $\singleton{\gamma_n}$ that tends to $\infty$ contains a subsequence %
+In conclusion, every sequence of scalars $\gamma_n$ such that $\gamma_n \to \infty$ contains a subsequence%
 $\singleton{\gamma_{n_k}}$ that causes $\singleton{\gamma_{n_k}f_{n_k}}$ to diverge. This is (b). 
 \end{proof}
 %

chapter_1/1_7.tex

@@ -16,23 +16,19 @@
 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
 % A. Assumptions.
 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
-Take $E_n$ and $V_n$ such that %
+Take a sequence of open sets $\singleton{V_n}$ where %
 %
 \renewcommand{\labelenumi}{(\roman{enumi})} 
 \begin{enumerate}
- \item $X = V_n \cup \overline{E}_n$ \quad ($V_n \cap \overline{E}_n = \emptyset)$;
- \item $X = \overline{V}_n$; 
+ \item $X = V_n \cup \closure{E}_n$ \quad ($V_n \cap \overline{E}_n = \emptyset)$;
+ \item $X = \closure{V_n}$; 
  \item $ X\setminus Y = \bigcup_{n=1}^\infty E_n$
 \end{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})} 
 %
-for all positive integers $n$. First, let $x$ be an arbitrary element of $X$. We note that $x + V_n$ is open and dense 
-as well (because%
-\footnote{
- This is also a special case of \citeresultFA{1.3 (b)}, since %
- $X = x + X \subset \overline{x + V_n}$.
-}
-the translation by $x$ is a homeomorphism of $X$ onto $X$).
+for all $n$. Fix $x \in X$. Then $x + V_n$ is also open and dense because the translation by $x$ is a %
+self-homeomorphism. Alternatively, observe that $X = x + X \subset \closure{x + V_n}$, %
+which is is a special case of \citeresultFA{1.3 (b)}. %
 %
 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
 % B. Where Baire's theorem is involved.

chapter_2/2_15.tex

@@ -100,7 +100,7 @@ \subsection{Equinumerosity}%
 \section{Topological vector spaces}
 \subsection{Scalar field}%
 $\C$ extends $\R$, which implies that a property\eg linearity, that is true on $\C$ is also true on $\R$. %
-The complex case is then a \textit{special case} of the real one. Sometimes, this restriction can be significant. %
+The complex case is then a \textit{special case} of the real case. Sometimes, this restriction can be significant. %
 Nevertheless, the standard scalar field is $\C$, which means that considering $\R$ instead of $\C$ makes no difference, %
 unless stated otherwise. %
 %
@@ -125,7 +125,7 @@ \subsection{Vector space bases}\label{notations: vector spaces: vector space bas
  \item all bases to have the same cardinal, which is called the {\it dimension} of $X$ and is denoted as $\dim(X)$. %
 \end{enumerate}
 %
-We now come to the finite-dimensional case. Note that the $0$-dimensional case is the degenerate case $B=\emptyset$, %
+We now turn to the finite-dimensional case. Note that the $0$-dimensional case is the degenerate case $B=\emptyset$, %
 which is equivalent to $X=\singleton{0}$. Our first step is to study $\C^n$ ($n>0$) the standard $n$-dimensional vector space.
 %
 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
@@ -189,7 +189,7 @@ \subsubsection{Topology of a finite-dimensional vector space}
  \norm{y} \leq C \norma{2}{z} \quad \bigl((z, y) \in f\bigr), 
 \end{align}
 %
-because $f$ is continuous. Now pick an $n$-dimensional topological vector space $W$, then reiterate the same reasoning, %
+because $f$ is continuous. Now pick an $n$-dimensional topological vector space $W$, then repeat the same reasoning, %
 first with $g: \C^n \to W$, next with $h = g\circ f^{\,\minus 1}$ playing the role of $f$. This establishes that the 
 homeomorphism $h$ maps $Y$ onto $W$ and that $W$ is normable as well. It is now clear that the following assertions are 
 equivalent in the finite-dimensional context, %