X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo-cone.tex;h=b53e23918d36c644d9ead02e35574d9adbdcc2a6;hb=21d18c2a88952e968b02c838abc5cb61dff960c9;hp=1199d4f55be54130d2660f7814f9f857d8590853;hpb=ad5dafa4ee98c4729b33bebc1817c09cae1b881a;p=mjotex.git

diff --git a/mjo-cone.tex b/mjo-cone.tex
index 1199d4f..b53e239 100644
--- a/mjo-cone.tex
+++ b/mjo-cone.tex
@@ -9,12 +9,16 @@
 
 \input{mjo-common}
 
+% The dual of a subset of an inner-product space; always a closed
+% convex cone.
+\newcommand*{\dual}[1]{ #1^{*} }
+
 %
 % Common cones.
 %
 
 % The nonnegative orthant in the given number of dimensions.
-\newcommand*{\Rnplus}[1][n]{ \mathbb{R}^{{#1}}_{+} }
+\newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
 
 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
 \newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
@@ -27,9 +31,25 @@
 
 
 %
-% Some collections of linear operators
+% Some collections of linear operators.
 %
 
+% The set of all positive operators on its argument. This uses the
+% same magic as \boundedops to accept either one or two arguments. If
+% one argument is given, the domain and codomain are equal and the
+% positive operators fix a subset of that space. When two arguments
+% are given, the positive operators send the first argument to a
+% subset of the second.
+\newcommand*{\posops}[2][]{
+  \pi\of{ {#2}
+    \if\relax\detokenize{#1}\relax
+      {}%
+    \else
+      {,{#1}}%
+    \fi
+  }
+}
+
 % The set of all S-operators on its argument.
 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
 
@@ -45,10 +65,10 @@
 %
 
 % Standard cone inequalities.
-\newcommand*{\gek}{ \succcurlyeq }
-\newcommand*{\gtk}{ \succ }
-\newcommand*{\lek}{ \preccurlyeq }
-\newcommand*{\ltk}{ \prec }
+\newcommand*{\gek}{\succcurlyeq}
+\newcommand*{\gtk}{\succ}
+\newcommand*{\lek}{\preccurlyeq}
+\newcommand*{\ltk}{\prec}
 
 
 % Starred versions of the cone inequality operators.