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

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -441,15 +441,15 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
 \end{proof}
 
 At this point it may be tempting to think that we could now define the metric
-of \(H^1(I, M)\) in a fiber-wise basis via the identification \(T_\gamma
-H^1(\gamma^* TM) \cong H^1(\gamma^* TM)\). In a very real sense this is what we
-are about to do, but unfortunately there are still technicalities in our way.
-The issue we face is that proposition~\ref{thm:h0-bundle-is-complete} only
-applies for \emph{smooth} vector bundles \(E \to I\), which may not be the case
-for \(E = \gamma^* TM\) if \(\gamma \in H^1(I, M)\) lies outside of
-\({C'}^\infty(I, M)\). In fact, neither \(\langle X , Y \rangle_0\) nor
-\(\langle \, , \rangle_1\) are defined \emph{a priori} for \(X, Y \in
-H^0(\gamma^* TM)\) with \(\gamma \notin {C'}^\infty(I, M)\).
+of \(H^1(I, M)\) in a fiber-wise basis via the identification \(T_\gamma H^1(I,
+M) \cong H^1(\gamma^* TM)\). In a very real sense this is what we are about to
+do, but unfortunately there are still technicalities in our way. The issue we
+face is that proposition~\ref{thm:h0-bundle-is-complete} only applies for
+\emph{smooth} vector bundles \(E \to I\), which may not be the case for \(E =
+\gamma^* TM\) if \(\gamma \in H^1(I, M)\) lies outside of \({C'}^\infty(I,
+M)\). In fact, neither \(\langle X , Y \rangle_0\) nor \(\langle \, ,
+\rangle_1\) are defined \emph{a priori} for \(X, Y \in H^0(\gamma^* TM)\) with
+\(\gamma \notin {C'}^\infty(I, M)\).
 
 Nevertheless, we can get around this limitation by extending the metric
 \(\langle \, , \rangle_0\) and the covariant derivative \(\frac\nabla\dt =