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\centerline{\textbf{ANNEXE}}\vva
\centerline{\textbf{PARTIES\hb\ TOTALEMENT\hb\ BORNEES\hb\ DE}\: $\mbof{\LL^1}$}\vvcd

Soit\hb\ $V$\ha\ un espace de Banach\vb

\underline{D\e finition}\dd Une partie de\hb\ $V$\ha\ est\, \underline{born\e e}\,
ssi elle est incluse \ac\ une boule de\ha\ $V$\rr.\vva

\underline{D\e finition}\dd Une partie de\hb\ $V$\ha\ est\, \underline{totalement born\e e}\, ssi\:
$\all\,\eps>0$\: elle est incluse \ac\ une r\e union\, \underline{finie}\, de boules de\hb\ $V$\ha\
de rayon\, $\eps$\va

Une partie totalement born\e e est \e videmment born\e e\vb

\underline{Th\e or\ee me fondamental}\dd Une partie\, \underline{born\e e}\, $A$\, de\, $\LL^1$\hb\ est\,
\underline{totalement born\e e}\: ssi\: $\all\,\eps>0$\: il existe\: $\del>0$\: tel que\vvb
\hh$1\hw^\circ)$\,\hh\fbox{\:$\all\ha\tdf\in A\ \dis\int_a^{\hb a\,+\,\del}\hs{-4mm}\vtdf\ha(x)\:d\hb x\hw
\le\ha\eps$\,}\quad et\quad\fbox{\:$\all\ha\tdf\in A\ \dis\int_{b\,-\ha\del}^{\hb b}\rs\vtdf\ha(x)\:d\hb x\hw
\le\ha\eps$\,}\vvb
\hh$2\hb^\circ)$\,\hh\fbox{\:$\all\ha\tdf\in A\ha\hh\all\,h\in\ha\EE\ha\hh\all\:u\in\uu\ha\
\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\hs{-3mm}\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha\tdf\ha(x)\,
d\hb x\,\right|\le\ha\eps\,\|\ha h\ha\|$\ha}\ha\vbc

\Dem\vva
a)\hh$\imp$\dd Soient\, $\tdf\in\ha\LL^1$\, et\, $h\in\ha\EE$\pv soit\, $\eps>0$\, et soit\, $g\in\CC$\,
tel que\, $\bnone{\ha\tdf\rr-g\ha}\le\eps$\pv\vvw
on a\hh$\all\:u\in\uu$\vvab
$\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha\tdf\ha(x)\,
d\hb x\ha-\rr\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha g\ha(x)\,
d\hb x\,\right|$\vva
$=\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha
\big[\ha\tdf\ha(x)-g\ha(x)\ha\big]\,d\hb x\ha\right|\le\ha2\:\|\ha h\ha\|\:\bnone{\ha\tdf\rr-g\ha}
\le\ha2\:\eps\:\|\ha h\ha\|$\pv de plus\vvaab

$\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha
g\ha(x)\,d\hb x\,\right|=\left|\ha\dis\int_{a\,+\,\del\,+\,u}^{\hb b\,-\ha\del\,+\,u}\rs h\ha(x)\,
g\ha(x\re-\re u)\,d\hb x\ha-\rr\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(x)\,g\ha(x)\,d\hb x\,\right|$\vva
$=\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs h\ha(x)\big[\hb g\ha(x\re-\re u)\re-g\ha(x)\ha\big]\,
d\hb x\,\right|\hb
+\hb\left|\ha\dis\int_{a\,+\,\del}^{\hb a\,+\,\del\,+\,u}\rs h\ha(x)\hb g\ha(x\re-\re u)\,d\hb x\,\right|$\vva
$+\,\left|\ha\dis\int_{b\,-\ha\del\,+\,u}^{\hb b\,-\ha\del}\rs h\ha(x)\hb g\ha(x\re-\re u)\,d\hb x\,\right|
\le\ha\|\ha h\ha\|\rr\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big|\hb g\ha(x\re-\re u)\re-g\ha(x)\ha\big|\,
d\hb x\,+\:2\;\del\,\|\ha g\ha\|\ha\|\ha h\ha\|$\pv\vvab

soit\hh$0<\del<\dis\frac{\eps}{\|\ha g\ha\|}$\hh tel que\, $\Big[\,|\ha x_1-x\ha_2\ha|\le\ha\del\,\imp\big|\hb
g\ha(x_1)\re-g\ha(x\ha_2)\ha\big|\le\hb\dis\frac{\eps}{b-a}\,\Big]$\ha;\;\vva
alors\,\hh$\all\:u\in\uu\,\hh\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs
\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha g\ha(x)\,d\hb x\,\right|\le\ha3\:\eps\,\|\ha h\ha\|$\ha\vg\vva
donc\,\hh$\all\:u\in\uu\,\hh\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs
\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha\tdf\ha(x)\,d\hb x\,\right|\le\ha5\:\eps\,\|\ha h\ha\|$\vab

\newpage
Choisissons dans\, $\LL^1$\, un ensemble fini de boules\vg de centres\, $\tdf_1\vv\tdf_1\ldots\tdf_n$\, et de
rayon\, $\eps$\vg recouvrant\, $A$\pv on peut choisir en vertu de ce qui pr\e c\ee de\, $\del>0$\, tel que\vva
$\all\,i\in[[\ha1\ha,\ha n\hb]]\,\hh\all\,h\in\ha\EE\,\hh\all\:u\in\uu\,\
\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha\tdf_i\ha(x)\,
d\hb x\,\right|\le\ha\eps\,\|\ha h\ha\|$\pv\vvaab

en r\e duisant \e ventuellement\, $\del$\, on peut exiger en outre\vva
$\all\,i\in[[\ha1\ha,\ha n\hb]]\hh\dis\int_a^{\hb a\,+\,\del}\hs{-4mm}\big|\ha\tdf_i\hw\big|\ha(x)\:d\hb x\hw
\le\ha\eps$\quad et\hh$\dis\int_{b\,-\ha\del}^{\hb b}\rs\big|\ha\tdf_i\hw\big|\ha(x)\:d\hb x\hw
\le\ha\eps$\ha\vab

Soit\, $\tdf\in A$\: et soit\, $i\in[[\ha1\ha,\ha n\hb]]$\, tel que\,
$\none{\ha\tdf-\tdf_i\ha}\le\ha\eps$\pv\vvab
alors\,\hh$\all\,h\in\ha\EE\hh\all\:u\in\uu\ha\ \left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs
\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha\tdf\ha(x)\,d\hb x\,\right|$\vva
$\le\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha\tdf_i\ha(x)\,
d\hb x\,\right|+\left|\ha\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rs\big[\hb h\ha(x+u)\re-h\ha(x)\ha\big]\ha
\big[\tdf\ha(x)-\tdf_i\ha(x)\big]\ha d\hb x\,\right|$\vvab
$\le\ha\eps\,\|\ha h\ha\|+2\:\eps\,\|\ha h\ha\|=\ha3\:\eps\,\|\ha h\ha\|$\vab

De plus on a\,\hh$\dis\int_a^{\hb a\,+\,\del}\hs{-4mm}\vtdf\ha(x)\:d\hb x\ha
\le\re\int_a^{\hb a\,+\,\del}\hs{-4mm}\big|\ha\tdf_i\hw\big|\ha(x)\:d\hb x\,
+\int_a^{\hb a\,+\,\del}\hs{-4mm}\big|\ha\tdf-\tdf_i\hw\big|\ha(x)\:d\hb x\ha\le\ha2\:\eps$\vc

b)\hh$\pmi$\dd Soit\, $\phi$\, la fonction\, $\R\to\R^+$\, dont le graphe est un
triangle isoc\ee le\vvw de base\hb\ $[\ha-\hb\del\hb,\ha\del\ha]$\, et de hauteur\, $1/\del$\pv
on a donc\ha\ $\dis\int_{\re-\ha\del}^{\,\del}\phi\hb(u)\,d\hb u=1$\hb.\vvw
$\all\ha\tdf\in\LL^1$\, d\e finissons\, la fonction de\, $\CC$\vvwa
\hs{35mm}$\tdf\re*\phi\::\hb\ab\to\R\,:\,x\tto\rs\dis\int_a^{\hb b}\phi\hb(x\re-u)\hb\tdf\ha(u)\:d\hb u$\vwa

On a\,\hh$\all\,h\in\ha\EE\,\ \dis\int_a^{\hb b}(\tdf\re*\phi)\ha(x)\,h\ha(x)\:d \hb x
=\rs\dis\int_a^{\hb b}\left[\int_a^{\hb b}\phi\hb(x\re-u)\,h\ha(x)\:d\hb x\ha\right]\rr\tdf(u)\:d\hb u$\vva
$=\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\left[\int_a^{\hb b}\phi\hb(x\re-u)\,h\ha(x)\:d\hb x\ha\right]\rr
\tdf(u)\:d\hb u\,+\re\underbrace{\dis\int_a^{\hb a\,+\,\del}\rs\rs\left[\ha\ldots\ha\right]\tdf(u)\:d\hb u}
_{K_1}\,+\re\underbrace{\dis\int_{b\,-\ha\del}^{\hb b}\rs\left[\ha\ldots\ha\right]\tdf(u)\:d\hb u}
_{L_{\hw1}}$\ha\vwa

De plus\vvab$\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\left[\int_a^{\hb b}\phi\hb(x\re-u)\,h\ha(x)\:d\hb x\ha
\right]\rr\tdf(u)\:d\hb u=\rs\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rr\left[\int_{\hb u\ha-\ha\del}
^{\hb u\,+\,\del}\phi\hb(x\re-u)\,h\ha(x)\:d\hb x\ha\right]\rr\tdf(u)\:d\hb u$\vva
$=\rs\dis\int_{a\,+\,\del}^{\hb b\,-\ha\del}\rr\left[\int_{\re-\ha\del}^{\,\del}\phi\hb(v)\,h\ha(v+u)\:
d\hb v\ha\right]\rr\tdf(u)\:d\hb u=\rs\dis\int_{\re-\ha\del}^{\,\del}
\left[\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(u+v)\hb\tdf(u)\:d\hb u\ha\right]\rr\phi\hb(v)\:d\hb v$\vvbc

On a aussi\vvab$\dis\int_a^{\hb b}h\ha(x)\ha\tdf\ha(x)\,d\hb x
=\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(x)\ha\tdf\ha(x)\,d\hb x\,
+\re\underbrace{\int_a^{\hb a\,+\,\del}h\ha(x)\ha\tdf\ha(x)\,d\hb x}_{K_2}\,
+\re\underbrace{\int_{b\,-\ha\del}^{\hb b}h\ha(x)\ha\tdf\ha(x)\,d\hb x}_{L_2}$\vw

\newpage
On peut donc \e crire\,\hh$\all\ha\tdf\in A\,\hh\all\,h\in\ha\EE\,\hh\left|\ha\dis\int_a^{\hb b}h\ha(x)\ha
\big[\ha(\tdf\re*\phi)\ha(x)\re-\re\tdf\ha(x)\ha\big]\hb d\hb x\,\right|$\vva
$\le\left|\ha\dis\int_{\re-\ha\del}^{\,\del}\left[\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(u+v)\hb\tdf(u)\:
d\hb u\ha\right]\rr\phi\hb(v)\:d\hb v\ha-\rr\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(x)\ha\tdf\ha(x)\,
d\hb x\,\right|\:+\,M$\vvb
avec\hh$M=|K_1|\hb+\hb|L\ha_1|\hb+\hb|K_2|\hb+\hb|L\ha_2|$\vb

Or on a\,\hh$\left|\ha\dis\int_{\re-\ha\del}^{\,\del}\left[\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(u+v)\hb
\tdf(u)\:d\hb u\ha\right]\rr\phi\hb(v)\:d\hb v\ha-\rr\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(x)\ha
\tdf\ha(x)\,d\hb x\,\right|$\vva
$=\left|\ha\dis\int_{\re-\ha\del}^{\,\del}\left[\int_{a\,+\,\del}^{\hb b\,-\ha\del}h\ha(u+v)\hb\tdf(u)\:
d\hb u\ha\right]\rr\phi\hb(v)\:d\hb v\ha-\rr\int_{\re-\ha\del}^{\,\del}\left[\int_{a\,+\,\del}
^{\hb b\,-\ha\del}h\ha(x)\ha\tdf\ha(x)\,d\hb x\ha\right]\rr\phi\hb(v)\:d\hb v\,\right|$\vva

$=\left|\ha\dis\int_{\re-\ha\del}^{\,\del}\left[\int_{a\,+\,\del}^{\hb b\,-\ha\del}
\big[h\ha(x+v)\re-h\ha(x)\big]\ha\tdf\ha(x)\,d\hb x\right]\rr\phi\hb(v)\:d\hb v\,\right|$\vva
$\le\dis\int_{\re-\ha\del}^{\,\del}\left|\ha\int_{a\,+\,\del}^{\hb b\,-\ha\del}\big[\hb h\ha(x+v)\re
-h\ha(x)\ha\big]\ha\tdf\ha(x)\,d\hb x\,\right|\re\phi\hb(v)\:d\hb v\ha\le\ha\eps\:\|\ha h\ha\|$\quad
en vertu de la condition\, $2\hb^\circ)$\vb

Par ailleurs en vertu de la condition\, $1\hw^\circ)$\, on a clairement\hh$M\le\ha4\:\eps\:\|\ha h\ha\|$\pv
on en d\e duit\vvwa$\all\ha\tdf\in A\,\ \left|\ha\dis\int_a^{\hb b}h\ha(x)\ha\big[\ha(\tdf\re*\phi)\ha(x)\re
-\re\tdf\ha(x)\ha\big]\hb d\hb x\,\right|\le\ha5\:\eps\:\|\ha h\ha\|$\ha\vg\ha donc\hh
\fbox{\hb$\none{\ha\tdf\re*\phi\ha-\tdf\ha}\le\ha5\:\eps$\hb}\ha\pv\vvwa
or\ha\hh$\all\ha\tdf\in A\ha\hh\tdf\re*\phi\in\ha\CC$\pv montrons que\, $A*\phi\ha\sub\hb\CC$\;
est totalement born\e\ dans\, $\CC$\ha,\, \cad\vvw d'apr\ee s le th\e or\ee me d'Ascoli\ha:\vvab

\quad* born\e\dd$\big|\ha(\tdf\re*\phi)\ha(x)\ha\big|=\left|\ha\dis\int_a^{\hb b}\phi\hb(x\re-u)\ha\tdf(u)\,
d\hb u\,\right|\le\ha\dis\inv{\del}\;\none{\ha\tdf\ha}$\vg\vva
\quad* \e quicontinu\dd$\big|\ha(\tdf\re*\phi)\ha(x)-(\tdf\re*\phi)\ha(y)\ha\big|=\left|\ha\dis\int_a^{\hb b}
\big[\ha\phi\hb(x\re-u)\re-\phi\hb(y\re-u)\hw\big]\ha\tdf(u)\,d\hb u\,\right|$\\
\hs{36mm}$\le\ha\dis\inv{\del\ha^2}\;\none{\ha\tdf\ha}\,|\ha x\re-y\ha|$\va

$A*\phi$\; peut donc \^etre recouvert par un ensemble fini de boules de\, $\CC$\, de rayon\,
$\dis\frac{\eps}{b-a}$\vg elles-m\^emes incluses aux boules de\, $\LL^1$\ha\ de m\^eme centre
et de rayon\, $\eps$\pv on peut donc recouvrir\, $A$\; par l'ensemble des boules de\, $\LL^1$\ha\
de m\^eme centre que les pr\e c\e dentes\ha\ et\\ de rayon\, $6\:\eps$\vb

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