Update documentation

_sources/abstracttheory/infsup.ipynb

@@ -39,7 +39,7 @@
     "\\| B u \\|_{W^\\ast} \\geq \\beta_1 \\| u \\|_V.\n",
     "$$\n",
     "\n",
-    "We immediately obtain that $B$ is one to one, since\n",
+    "We immediately obtain that $B$ is one to one (aka injective), since\n",
     "\n",
     "$$\n",
     "B u = 0 \\Rightarrow u = 0\n",
@@ -52,7 +52,7 @@
     "converges to some $u \\in V$. By continuity of $B$, the sequence $B u^n$ converges to $B u \\in W^\\ast$.\n",
     " $\\Box$\n",
     "\n",
-    "The inf-sup condition does not imply that $B$ is onto $W^\\ast$. To insure that, we can pose an inf-sup condition the other way around:\n",
+    "The inf-sup condition does not imply that $B$ is onto $W^\\ast$. To insure that, we can pose an inf-sup condition the other way around (aka surjective):\n",
     "\n",
     "$$\n",
     "\\inf_{v \\in W \\atop v \\neq 0} \\sup_{u \\in V \\atop u \\neq 0}\n",

abstracttheory/infsup.html

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@@ -507,7 +507,7 @@ <h1><span class="section-number">11. </span>Inf-sup stable variational problems<
 \[
 \| B u \|_{W^\ast} \geq \beta_1 \| u \|_V.
 \]</div>
-<p>We immediately obtain that <span class="math notranslate nohighlight">\(B\)</span> is one to one, since</p>
+<p>We immediately obtain that <span class="math notranslate nohighlight">\(B\)</span> is one to one (aka injective), since</p>
 <div class="math notranslate nohighlight">
 \[
 B u = 0 \Rightarrow u = 0
@@ -519,7 +519,7 @@ <h1><span class="section-number">11. </span>Inf-sup stable variational problems<
 <p><em>Proof:</em> Let <span class="math notranslate nohighlight">\(B u^n\)</span> be a Cauchy sequence in <span class="math notranslate nohighlight">\(W^\ast\)</span>. From <span class="math notranslate nohighlight">\(\| Bu \| \geq \beta_1 \| u \|\)</span> we conclude that also <span class="math notranslate nohighlight">\(u^n\)</span> is Cauchy in <span class="math notranslate nohighlight">\(V\)</span>. Since <span class="math notranslate nohighlight">\(V\)</span> is complete, <span class="math notranslate nohighlight">\(u_n\)</span>
 converges to some <span class="math notranslate nohighlight">\(u \in V\)</span>. By continuity of <span class="math notranslate nohighlight">\(B\)</span>, the sequence <span class="math notranslate nohighlight">\(B u^n\)</span> converges to <span class="math notranslate nohighlight">\(B u \in W^\ast\)</span>.
 <span class="math notranslate nohighlight">\(\Box\)</span></p>
-<p>The inf-sup condition does not imply that <span class="math notranslate nohighlight">\(B\)</span> is onto <span class="math notranslate nohighlight">\(W^\ast\)</span>. To insure that, we can pose an inf-sup condition the other way around:</p>
+<p>The inf-sup condition does not imply that <span class="math notranslate nohighlight">\(B\)</span> is onto <span class="math notranslate nohighlight">\(W^\ast\)</span>. To insure that, we can pose an inf-sup condition the other way around (aka surjective):</p>
 <div class="math notranslate nohighlight">
 \[
 \inf_{v \in W \atop v \neq 0} \sup_{u \in V \atop u \neq 0}
@@ -636,8 +636,8 @@ <h2><span class="section-number">11.1. </span>Approximation of inf-sup stable va
 of the Babuška-Aziz theorem. Furthermore,
 <span class="math notranslate nohighlight">\(B(\cdot,\cdot)\)</span> fulfills the discrete inf-sup condition with bound <span class="math notranslate nohighlight">\(\beta_{1h}\)</span>.
 Then there holds the quasi-optimal error estimate</p>
-<div class="amsmath math notranslate nohighlight" id="equation-e14b1323-ff75-4015-bcf6-b7df75032693">
-<span class="eqno">()<a class="headerlink" href="#equation-e14b1323-ff75-4015-bcf6-b7df75032693" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-908c16fa-2431-4975-8643-bd4697baf652">
+<span class="eqno">()<a class="headerlink" href="#equation-908c16fa-2431-4975-8643-bd4697baf652" title="Permalink to this equation">#</a></span>\[\begin{equation}
   \| u - u_h \| \leq (1 + \beta_2 / \beta_{1h}) \inf_{v_h \in V_h} \| u - v_h \|
   \end{equation}\]</div>
 </div></blockquote>