This commit is contained in:
@@ -7,7 +7,7 @@
|
||||
\begin{enumerate}
|
||||
\item[(LO1)] For any $x, y, z \in E$ with $x \le y$, $x + z \le y + z$.
|
||||
\item[(LO2)] For any $x, y \in E$ and $\lambda > 0$, $x \le y$ implies that $\lambda x \le \lambda y$.
|
||||
\end\{enumerate\}
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\end{definition}
|
||||
@@ -18,7 +18,7 @@
|
||||
\begin{enumerate}
|
||||
\item $\sup(A + B) = \sup(A) + \sup(B)$.
|
||||
\item $\sup(A) = -\inf (-A)$
|
||||
\end\{enumerate\}
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\end{proposition}
|
||||
@@ -105,13 +105,13 @@
|
||||
\begin{enumerate}
|
||||
\item[(3)] $|\lambda x| = |\lambda| \cdot |x|$
|
||||
\item[(4)] $|x + y| \le |x| + |y|$.
|
||||
\end\{enumerate\}
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
Finally, for any $x, y \in E$ with $x, y \ge 0$,
|
||||
\begin{enumerate}
|
||||
\item[(5)] $[0, x] + [0, y] = [0, x + y]$.
|
||||
\end\{enumerate\}
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\end{proposition}
|
||||
@@ -179,7 +179,7 @@
|
||||
\begin{enumerate}
|
||||
\item For any $x \in C$ and $\lambda \in \real$ with $\lambda \ge 0$, $\phi(\lambda x) = \lambda \phi(x)$.
|
||||
\item For any $x, y \in C$, $\phi(x + y) = \phi(x) + \phi(y)$.
|
||||
\end\{enumerate\}
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
then the mapping
|
||||
@@ -223,7 +223,7 @@
|
||||
\[
|
||||
|\phi|(x) = \sup\bracs{\phi(y)|y \in E, |y| \le x}
|
||||
\]
|
||||
\end\{enumerate\}
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\end{proposition}
|
||||
|
||||
Reference in New Issue
Block a user