global-analysis-and-the-banach-manifold-of-class-h1-curvers

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -107,10 +107,10 @@ spaces, and consequently generalize smooth manifolds to spaces modeled after
 Banach spaces. We begin by the former.
 
 \begin{definition}
-  Let \(V\) and \(W\) be Banach manifolds and \(U \subset V\) be an open
-  subset. A continuous map \(f : U \to W\) is called \emph{differentiable at
-  \(p \in U\)} if there exists a continuous linear operator \(d f_p \in
-  \mathcal{L}(V, W)\) such that
+  Let \(V\) and \(W\) be Banach spaces and \(U \subset V\) be an open subset. A
+  continuous map \(f : U \to W\) is called \emph{differentiable at \(p \in U\)}
+  if there exists a continuous linear operator \(d f_p \in \mathcal{L}(V, W)\)
+  such that
   \[
     \frac{\norm{f(p + h) - f(p) - d f_p h}}{\norm{h}} \to 0
   \]

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -22,11 +22,10 @@ Finally, we may define\dots
 \begin{definition}
   Given an \(n\)-dimensional manifold \(M\), a continuous curve \(\gamma : I
   \to M\) is called \emph{a class \(H^1\)} curve if \(\varphi_i \circ \gamma :
-  J \to \RR^n\) for any chart \(\varphi_i : U_i \subset M \to \RR^n\) and
-  \(J \subset I\) a maximal subinterval where \(\varphi_i \circ \gamma\) is
-  defined -- i.e. if \(\gamma\) can be locally expressed as a class \(H^1\)
-  curve in terms of the charts of \(M\). We'll denote by \(H^1(I, M)\) of all
-  class \(H^1\) curves \(I \to M\).
+  J \to \RR^n\) is a class \(H^1\) curve for any chart \(\varphi_i : U_i
+  \subset M \to \RR^n\) -- i.e. if \(\gamma\) can be locally expressed as a
+  class \(H^1\) curve in terms of the charts of \(M\). We'll denote by \(H^1(I,
+  M)\) the set of all class \(H^1\) curves \(I \to M\).
 \end{definition}
 
 \begin{note}
@@ -67,7 +66,7 @@ values of \(n\), as shown in figure~\ref{fig:step-curves}: clearly \(\gamma_n
 \to \gamma\) in the uniform topology, but \(L(\gamma_n) = 2\) does not
 approach \(L(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
 
-\begin{figure}[h]\label{fig:step-curves}
+\begin{figure}[h]
   \centering
   \begin{tikzpicture}
     \draw (4, 1) -- (1, 4);
@@ -87,6 +86,7 @@ approach \(L(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
   \caption{A diagonal line representing the curve $\gamma$ overlaps a
   staircase-like curve $\gamma_n$, whose steps measure $\sfrac{1}{n}$ in
   width and height.}
+  \label{fig:step-curves}
 \end{figure}
 
 The issue with this particular example is that while \(\gamma_n \to \gamma\)