UPDATE: started reviewing chapter 6.

Author: gitcordier

FA_DM.pdf

@@ -130,7 +130,7 @@
 \newcommand{\function}[1]{\mathtt{#1}}
 \newcommand{\relation}[2]{{#1}_{#2}}
 \newcommand{\f}[2]{#1(#2)}
-\DeclareMathOperator\supp{supp}
+
 
 % Sets
 %\renewcommand{\notin}{\tiny{\not\in}}

FA_DM.tex

@@ -1,15 +1,15 @@
 {\CMUCS Suppose $f$ is a complex continuous function in $\R^n$, with compact support. \!\!Prove that $\psi P_j\to \!f$ uniformly on $\R^n$, for some $\psi\in \mathscr{D}$ and for some sequence $\{P_j\,\}$ of polynomials.}
-\paragraph{PROOF.} According to 1.16, $\Omega$ is union of a compact sets sequence $\{K_{i\,}\}$ and $\text{supp} (f\,)$ lies in some $K= K_{i\,}$, so that $f\,$ is embedded in $\mathscr{D}(\Omega)\,$. We can apply [1.10] to ensure that $\Omega$ encloses a compact set $S=\overline{K +B(\eps)}$ for sufficiently small $\eps>0\,$.\\
+\paragraph{PROOF.} According to 1.16, $\Omega$ is union of a compact sets sequence $\{K_{i\,}\}$ and $\text{supp} (f\,)$ lies in some $K= K_{i\,}$, so that $f\,$ is embedded in $\mathscr{D}(\Omega)\,$. We can apply [1.10] to ensure that $\Omega$ encloses a compact set $S=\overline{K +B(\epsilon)}$ for sufficiently small $\epsilon>0\,$.\\
 \\
 One easily checks that the Stone-Weierstraß theorem [5.7] can be applied with the subalgebra $ \{ g\in C(S\,) :\, g \text{ is polynomial}\,\}$ of $C(S\,)\,$.
 There so exists a sequence $\{P_j:\, j\in \N\}$ of $\R[X_1,\dotsc,\, X_n]\,$ such that 
 \begin{align}\label{6_1_1}
 \sup_S \lvert\, f-P_{j\,} \rvert \underset{j\infty}{\longrightarrow} 0\quad.
 \end{align} 
- By [6.20], the open set $K +B(\eps)$ has a local partition of unity $\{\psi_i\}\subseteq \mathscr{D}(\Omega)\,$. Moreover, there exists an integer $l$ such that $\psi=\psi_1+\dotsb+\psi_l\,$ equals $1$ on $K\,$. Hence
+ By [6.20], the open set $K +B(\epsilon)$ has a local partition of unity $\{\psi_i\}\subseteq \mathscr{D}(\Omega)\,$. Moreover, there exists an integer $l$ such that $\psi=\psi_1+\dotsb+\psi_l\,$ equals $1$ on $K\,$. Hence
 \begin{align}\label{6_1_2}
 \| \, f - \psi P_{j\,}\|_\infty= \, \| \,\psi f - \psi P_{j\,}\|_\infty 
 = & \, \sup_S \lvert \, \psi f -\! \psi P_{j\,}\rvert \\
- \< & \, \sup_S \lvert \, f - \, P_{j\,} \rvert \, \overset{(\ref{6_1_1})}{\underset{j\infty}{\longrightarrow}} 0\quad .
+ \minuseq & \, \sup_S \lvert \, f - \, P_{j\,} \rvert \, \overset{(\ref{6_1_1})}{\underset{j\infty}{\longrightarrow}} 0\quad .
 \end{align} 
 \QED

chapter_6/6_1.tex

@@ -9,30 +9,30 @@
 \end{align}
 Now let $m$ range over $\{1,\, 2,\,3,\, \dots\}$ and set $W_{m,\, j\,}$ in $\mathscr{D}(\Omega)$ as follows
 \begin{align}\label{6_1_5a}
-D^{\,\moins \alpha} \phi \in \mathscr{D}(\Omega):\, D^{\,\alpha} D^{\,\moins \alpha} \phi= \phi\quad .
+D^{\,\minus \alpha} \phi \in \mathscr{D}(\Omega):\, D^{\,\alpha} D^{\,\minus \alpha} \phi= \phi\quad .
 \end{align}
 %\begin{align}\label{6_1_5c}
-%D^{\,\moins \alpha} \phi(x\,) \Def 
+%D^{\,\minus \alpha} \phi(x\,) \Def 
 %\underset{ \alpha_1 \text{ time(s)} }{
- % \underbrace{ \int_{\moins \infty}^{x_1} \dotsc\int_{\moins\infty}^{x_1}}
+ % \underbrace{ \int_{\minus \infty}^{x_1} \dotsc\int_{\minus\infty}^{x_1}}
 % }
 % \,\,\dotsc\,
 % \underset{ \alpha_n \text{ time(s)} }{
- % \underbrace{ \int_{\moins \infty}^{x_n} \dotsc\int_{\moins\infty}^{x_n}}
+ % \underbrace{ \int_{\minus \infty}^{x_n} \dotsc\int_{\minus\infty}^{x_n}}
  % } 
  % \,\, \phi \quad (x\in \R^n)\quad .
  % \end{align}
 \begin{align}\label{6_1_5}
-W_{m,\, j\,}(x\,)\Def D^{\,\moins (m,\dotsc,\,m)} ( \psi P_{(m,\dotsc, \, m),\, j\,})
+W_{m,\, j\,}(x\,)\Def D^{\,\minus (m,\dotsc,\,m)} ( \psi P_{(m,\dotsc, \, m),\, j\,})
  \end{align}
 By (\ref{6_1_4}), there exists a natural number $k(\!m)$ such that 
 \begin{align}\label{6_1_6}
-\|D^{(m,\dotsc,\, m)} (f -W_{m,\, j}) \|_\infty < 1/m \quad (j\> k(\!m) )\quad .
+\|D^{(m,\dotsc,\, m)} (f -W_{m,\, j}) \|_\infty < 1/m \quad (j\geq k(\!m) )\quad .
 \end{align}
  Assume without loss of generality that $S$ has diameter $1$, so that (\ref{6_1_6}) yields
 \begin{align}\label{6_1_7}
-\|D^{\, \lambda } (f -W_{m,\, k(\!m)} ) \|_\infty < 1/m \quad ( \lvert \lambda \rvert \< m )\quad ,
-%\max \{ \lvert D^{\, \lambda}(f- W_{m,\, k(\!m) })(x\,)\rvert :\, x\in S,\, \lvert \lambda\rvert \< m\,\} < 1/m \quad 
+\|D^{\, \lambda } (f -W_{m,\, k(\!m)} ) \|_\infty < 1/m \quad ( \lvert \lambda \rvert \leq m )\quad ,
+%\max \{ \lvert D^{\, \lambda}(f- W_{m,\, k(\!m) })(x\,)\rvert :\, x\in S,\, \lvert \lambda\rvert \leq m\,\} < 1/m \quad 
 \end{align}
 by the mean value theorem. In other words $(\text{remark that }\text{supp} (f -W_{m,\, k(\!m)}) \subseteq S\,)$, 
 \begin{align}\label{6_1_8}
@@ -46,19 +46,19 @@
 \\
 Choose $\delta $ in $\R_+$ and fetch any $W_{m,\, k(\!m)}$. Let $X$ be $(X_1,\dotsc,\, X_n)$ and express $P_{(m,\dotsc,\, m\,),\, k(\!m)}$ as 
 \begin{align}\label{6_1_10}
-P(X\,)=\sum_{\lvert \gamma \rvert \< d} p_\gamma \mdot X^{\,\gamma} \quad .
+P(X\,)=\sum_{\lvert \gamma \rvert \leq d} p_\gamma \cdot X^{\,\gamma} \quad .
 \end{align}
-Since $\overline{\Q}=\R$, $\Q[X\,]$ hosts some $\disp{Q(X\,)=\sum_{\lvert \gamma \rvert \< d} q_\gamma \mdot X^{\,\gamma}}$ such that $\lvert p_\gamma - q_\gamma\rvert < \delta\,$ for all $\gamma$. Thus,
+Since $\overline{\Q}=\R$, $\Q[X\,]$ hosts some $\{Q(X\,)=\sum_{\lvert \gamma \rvert \leq d} q_\gamma \cdot X^{\,\gamma}}$ such that $\lvert p_\gamma - q_\gamma\rvert < \delta\,$ for all $\gamma$. Thus,
 \begin{align}\label{6_1_11}
-\lvert P(x\,)-Q(x\,) \,\rvert \< \sum_{\lvert \gamma \rvert \< d} \lvert p_\gamma-q_\gamma\rvert\, \lvert x\,\rvert^{\lvert \gamma\rvert} \< \delta \sum_{l \< d} \binom{l+n-1}{n-1}\, \| \, x\, \|^{l}_\infty \quad (x\in \R^n)\quad .
+\lvert P(x\,)-Q(x\,) \,\rvert \leq \sum_{\lvert \gamma \rvert \leq d} \lvert p_\gamma-q_\gamma\rvert\, \lvert x\,\rvert^{\lvert \gamma\rvert} \leq \delta \sum_{l \leq d} \binom{l+n-1}{n-1}\, \| \, x\, \|^{l}_\infty \quad (x\in \R^n)\quad .
 \end{align}
 Since $S$ is bounded, we so obtain
 \begin{align}\label{6_1_12}
  \|\psi (P- Q) \|_\infty \in O(\delta) \quad .
 \end{align}
-Now define $\tilde{W}_m\,$ in terms of $Q$ as $W_{m,\, k(\!m)}$ was defined in terms of $P$, and consider the integrations made in (\ref{6_1_5}): each $D^{\,\lambda} \tilde{W}_m\,\, (\lvert \lambda \rvert\< m)$ can be obtained from some of them. So (\ref{6_1_12}) yields
+Now define $\tilde{W}_m\,$ in terms of $Q$ as $W_{m,\, k(\!m)}$ was defined in terms of $P$, and consider the integrations made in (\ref{6_1_5}): each $D^{\,\lambda} \tilde{W}_m\,\, (\lvert \lambda \rvert\leq m)$ can be obtained from some of them. So (\ref{6_1_12}) yields
 \begin{align}\label{6_1_13}
- \| D^{\,\lambda} (W_{m,\, k(\!m)} - \tilde{W}_m ) \|_\infty \in O(\delta)\quad (\lvert \lambda \rvert\< m) \quad . 
+ \| D^{\,\lambda} (W_{m,\, k(\!m)} - \tilde{W}_m ) \|_\infty \in O(\delta)\quad (\lvert \lambda \rvert\leq m) \quad . 
 \end{align}
 To be more specific, these $\lambda$'s only exist in finite amount, so the big O can be assumed to be the same for all them. Since $\delta$ was arbitrary, combining (\ref{6_1_9}) with (\ref{6_1_13}) establishes the density of the all $\tilde{W}_m$'s family $\tilde{W}$.\\
 \\

chapter_6/6_1_Mat.tex

@@ -2,7 +2,7 @@
 \renewcommand{\labelenumi}{(\alph{enumi})} 
 {\CMUCS
 \begin{enumerate}
-\item Suppose that $c_m=\exp\{\moins (m!)!\}$, $m=0,\, 1,\, 2,\, \dots\, $. Does the series
+\item Suppose that $c_m=\exp\{\minus (m!)!\}$, $m=0,\, 1,\, 2,\, \dots\, $. Does the series
 \begin{align*}{
 \sum_{m=0}^\infty c_m (D^m\phi)(0)
 }\end{align*}
@@ -17,11 +17,11 @@
 \\
 To do so we assume $\{\Lambda_j\}$ to converge to some $\Lambda$ of $\mathscr{D}'(\Omega)$ and we let $Q$ run through the compact sets of $\Omega\,$. Next, we define
 \begin{align}{\label{6_6_1}
-S(T,\, Q\,)\Def \{N\in \N, \, \exists C\in \R_+:\, \lvert T\phi\, \rvert \< C\, \| \phi \|_N \,\text{ for all }\phi \text{ of } \mathscr{D}_Q \}\quad (T\in \mathscr{D}(\Omega))\quad .
+S(T,\, Q\,)\Def \{N\in \N, \, \exists C\in \R_+:\, \lvert T\phi\, \rvert \leq C\, \| \phi \|_N \,\text{ for all }\phi \text{ of } \mathscr{D}_Q \}\quad (T\in \mathscr{D}(\Omega))\quad .
 }\end{align}
 Such subset of $\N$ has a minimum $\omega(T,\, Q)$. The following value
 \begin{align}{\label{6_6_2}
-\omega (T\,) \Def \max\{ \omega(T,\, Q\,): \, Q\subseteq \Omega\, , \,\, Q \text{ compact }\}\< \infty
+\omega (T\,) \Def \max\{ \omega(T,\, Q\,): \, Q\subseteq \Omega\, , \,\, Q \text{ compact }\}\leq \infty
 }\end{align}
 is then the order of $T$. Assume, to reach a contradiction, that, after passage to a subsequence,
 \begin{align}{\label{6_6_3}
@@ -30,11 +30,11 @@
 for some compact $Q=Q_j\,$. By (a) of [6.24], $Q_j$ cuts $\text{supp}\Lambda_j\,$, say in $p_j\,$. Since $K$ encloses $\text{supp}\Lambda_j\,$, $\{p_j\}$ tends, after passage to a subsequence, to some $p$ of $K\,$.
 Choose a positive scalar $r$ so that 
 \begin{align}{\label{6_6_5}
-\overline{B}(p,\, r)\Def \{ x\in \R^n:\, \lvert x-p\,\rvert\< r\}\subseteq \Omega \quad .
+\overline{B}(p,\, r)\Def \{ x\in \R^n:\, \lvert x-p\,\rvert\leq r\}\subseteq \Omega \quad .
 }\end{align}
 Such closed ball $\overline{B}(p,\, r)$ is a compact subset of $\Omega$. By (b) of [6.5] (which refers to [1.46]) $\mathscr{D}_{\overline{B}(p,\, r)}$ is then a Fréchet space. It now follows from [2.6] that $\{\Lambda_j\}$ is equicontinuous on $\mathscr{D}_{\overline{B}(p,\, r)}\,$. There so exists\footnote{For more details, see Exercise 2.3.} a nonnegative integer $N$ such that 
 \begin{align}{\label{6_6_6}
-\lvert \Lambda \phi \,\rvert \< C\, \| \phi \|_N \quad (\phi \in \mathscr{D}_{\overline{B}(p,\, r)})
+\lvert \Lambda \phi \,\rvert \leq C\, \| \phi \|_N \quad (\phi \in \mathscr{D}_{\overline{B}(p,\, r)})
 }\end{align}
 for some positive constant $C$. On the other hand, $\overline{B}(p,\, r)$ contains almost all the $p_j$'s. Hence
 \begin{align}{\label{6_6_7} 
@@ -45,12 +45,12 @@
 To prove (c), we introduce a sequence $\{x_m:\, m\in \Z\}$ of $ \Omega$ that has no limit point. Let $\{ \alpha_m:\, m\in \Z\}$ be in $\N$ and so define\footnote{As $\Omega=\R\,$, the case $\alpha_m= m$ is the ``counterpart" of the series of (a) and the case $(x_m,\, \alpha_m)= (m,\, 0)$ is the \textsl{Dirac comb}.}
 \begin{align}{
 \Lambda:\, \mathscr{D}(\Omega) \to & \, \C \qquad\qquad\qquad\qquad .\\
- \phi \mapsto & \, \sum_{m=\moins \infty}^\infty (D^{\,\alpha_m}\phi)(x_m) \nonumber
+ \phi \mapsto & \, \sum_{m=\minus \infty}^\infty (D^{\,\alpha_m}\phi)(x_m) \nonumber
 }\end{align}
 $\Lambda$ belongs to $\mathscr{D}'(\Omega)$, since $\{x_m\}$ has no limit point. Next, we easily check that
 \begin{align}{
 \Lambda_j:\, \mathscr{D}(\Omega) \to & \, \C \qquad\qquad\qquad\qquad(j\in \N)\\
- \phi \mapsto & \, \sum_{\lvert m\rvert \< j} (D^{\,\alpha_m}\phi)(x_m) \nonumber
+ \phi \mapsto & \, \sum_{\lvert m\rvert \leq j} (D^{\,\alpha_m}\phi)(x_m) \nonumber
 }\end{align}
 is also a distribution and that $\{\Lambda_j\}$ tends to $\Lambda$ in $\mathscr{D}'(\Omega)$. Nevertheless, no $\Lambda_j$'s can have common support because $\{x_m\}$ has no limit point. Our assumption can therefore be dropped.\QED
 

chapter_6/6_6.tex

@@ -1,13 +1,14 @@
 %!TEX root = /Volumes/HD_2/Rudin/Rudin_DM.tex
 \section{Exercise 1. Test functions are almost polynomial}
-%\input{\ROOT/chapter_6/6_1.tex}
+\input{\ROOT/chapter_6/6_1.tex}
+\input{\ROOT/chapter_6/6_1_Mat.tex}
 \setcounter{section}{5} 
 \section{Exercise 6. Around the supports of some distributions}
-%\input{\ROOT/chapter_6/6_6.tex}
+\input{\ROOT/chapter_6/6_6.tex}
 \setcounter{section}{8} 
 \section{Exercise 9. Convergence in $\mathscr{D}(\Omega)\,$ vs. convergence in $\mathscr{D}'(\Omega)\,$}
-%\input{\ROOT/chapter_6/6_9.tex}
-\setcounter{section}{16} 
-\section{Exercise 17. }
+\input{\ROOT/chapter_6/6_9.tex}
+%\setcounter{section}{16} 
+%\section{Exercise 17. }
 %\input{\ROOT/chapter_6/6_17.tex}