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contents/topology.tex : Tweaked topological space, metric and norm definition

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contents/topology.tex
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La topologie traite de l'étude des applications continues.
\langsection{Espaces topologique}{Topologic spaces}
\langsection{Espaces topologique}{Topological spaces}
A metric space is a set $E$ with a topology $\tau_E$ noted $(E,\tau_E)$.
\begin{definition_sq} \label {definition:topological_space}
\lang{Un espace topologique est un ensemble $E$ avec une topologie $\tau_E$ noté comme une paire $(E, \tau_E)$ vérifiant les axiomes suivants}%
{A topology space is a set $E$ with a topology $\tau_E$ noted as a pair $(E,\tau_E)$ satisfying the following axioms} :
\langsubsection{Axiomes}{Axioms}
\begin{itemize}
\item{$\{\emptyset, E\} \subseteq \tau_E$}
\item{Every union of open of $E$ is open, therefore in $\tau_E$ i.e. $\Union\limits_{F \in \powerset{E}}^{n \in \N^* \lor \infty} \in \tau_E$}
\item{Every finite intersection of open of $E$ is open, therefore in $\tau_E$ i.e. $\Intersection\limits_{F \in \powerset{E}}^{n \in \N^*} \in \tau_E$}
\end{itemize}
\begin{itemize}
\item{$\{\emptyset, E\} \subseteq \tau_E$}
\item{Every union of open of $E$ is open, therefore in $\tau_E$ i.e. $\Union\limits_{F \in \powerset{E}}^{n \in \N^* \lor \infty} \in \tau_E$}
\item{Every finite intersection of open of $E$ is open, therefore in $\tau_E$ i.e. $\Intersection\limits_{F \in \powerset{E}}^{n \in \N^*} \in \tau_E$}
\end{itemize}
\end{definition_sq}
\langsection{Espaces métrique}{Metric spaces}
\begin{definition_sq} \label{definition:metric_space}
A metric space is a set $E$ with a distance function $\function{d}{E^2}{\R_+}$ noted $(E,d)$ satisfaing the following axioms :
\lang{Un espace métrique est un ensemble $E$ avec une fonction de distance $\function{d}{E^2}{\R_+}$ notée comme une paire $(E, d)$ vérifiant les axiomes suivants}%
{A metric space is a set $E$ with a distance function $\function{d}{E^2}{\R_+}$ noted as a pair $(E, d)$ satisfying the following axioms} :
\begin{itemize}
\item{$\forall x,y \in E, d(x,y) = 0 \equivalence x = y$}
\item{Symetry: $\forall x,y \in E, d(x,y) = d(y,x)$}
\item{Triangular inegality: $\forall x,y,z \in E, d(x,y) \le d(x,z) + d(z,y)$}
\item{\lang{Non-dégénérescence}{Non-degenerative} : $\forall x,y \in E, d(x,y) = 0 \equivalence x = y$}
\item{\lang{Symétrie}{Symetry} : $\forall x,y \in E, d(x,y) = d(y,x)$}
\item{\lang{Inégalité triangulaire}{Triangular inegality} : $\forall x,y,z \in E, d(x,y) \le d(x,z) + d(z,y)$}
\end{itemize}
\end{definition_sq}
\langsubsection{Espaces vectoriels normés en dimension finie}{Vector spaces in finite dimensions}
Dans cette section, $E$ sera un $\R$-espace vectoriel.
\langsubsubsection{Normes}{Norms}
Une norme sur $E$ est une application continue qui vérifie certaines propriétés.
\begin{definition_sq}
Une norme sur $E$ est une application continue notée $\function{\norm{.}}{E}{\R_+}$ qui vérifie les axiomes suivants :
\smallskip
$\function{\norm{.}}{E}{\R_+}$
\langsubsubsubsection{Axiomes}{Axioms}
\begin{itemize}
\item{$\norm{x} = 0 \equivalence x = 0$}
\item{$\forall \lambda \in \R, \norm{\lambda x} = \abs{\lambda}\norm{x}$}
\item{$\forall(x,y) \in E, \norm{x + y} \le \norm{x} + \norm{y}$} (inégalité triangulaire)
\end{itemize}
\smallskip
\begin{itemize}
\item{Non-dégénérescence : $\norm{x} = 0 \equivalence x = 0$}
\item{Homothétie positive : $\forall \lambda \in \R, \norm{\lambda x} = \abs{\lambda}\norm{x}$}
\item{Inégalité triangulaire : $\forall(x,y) \in E, \norm{x + y} \le \norm{x} + \norm{y}$}
\end{itemize}
\end{definition_sq}
On appellera $(E,\norm{.})$ un \textbf{espace vectoriel normé}.
\langsubsubsubsection{Exemples}{Examples}
$n \in \N^*, E = \R^n$
Soit $n \in \N^*, E = \R^n$
\begin{itemize}
\item{$\norm{x}_1 = \sum\limits_{i = 1}^n \abs{x_i}$}