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

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -425,7 +425,7 @@ words, we'll show\dots
 
 \begin{theorem}\label{thm:h0-has-metric-extension}
   The vector bundle \(\coprod_{\gamma \in H^1(I, M)} H^0(\gamma^* TM) \to
-  H^1(I, M)\) admits a canonical Riemannian metric whose restriction the the
+  H^1(I, M)\) admits a canonical Riemannian metric whose restriction to the
   fibers \(H^0(\gamma^* TM) = \left.\coprod_{\eta \in H^1(I, M)} H^0(\eta^*
   TM)\right|_\gamma\) for \(\gamma \in {C'}^\infty(I, M)\) is given by
   \(\langle \, , \rangle_0\) as defined in
@@ -498,9 +498,9 @@ words, we'll show\dots
   simply by \(\frac\nabla\dt X\).
 \end{proposition}
 
-The proof of this last two propositions was deemed too technical to be included
-in here, but see proposition 2.3.16 and 2.3.18 of \cite{klingenberg}. We may
-now finally describe the canonical Riemannian metric of \(H^1(I, M)\).
+The proofs of this last two propositions were deemed too technical to be
+included in here, but see proposition 2.3.16 and 2.3.18 of \cite{klingenberg}.
+We may now finally describe the canonical Riemannian metric of \(H^1(I, M)\).
 
 \begin{definition}\label{def:h1-metric}
   Given \(\gamma \in H^1(I, M)\) and \(X, Y \in H^1(\gamma^* TM)\), let
@@ -514,11 +514,10 @@ now finally describe the canonical Riemannian metric of \(H^1(I, M)\).
 At this point it should be obvious that definition~\ref{def:h1-metric} does
 indeed endow \(H^1(I, M)\) with the structure of a Riemannian manifold: the
 inner products \(\langle \, , \rangle_1 : H^1(\gamma^* TM) \times H^1(\gamma^*
-TM) \to \RR\) may be glued together into a single positive-definite
-section \(\langle \, , \rangle_1 \in \Gamma( \Sym^2 \coprod_\gamma H^1(\gamma^*
-TM))\) -- whose smoothness follows from
-theorem~\ref{thm:h0-has-metric-extension},
+TM) \to \RR\) may be glued together into a single positive-definite section
+\(\langle \, , \rangle_1 \in \Gamma( \Sym^2 \coprod_\gamma H^1(\gamma^* TM))\)
+-- whose smoothness follows from theorem~\ref{thm:h0-has-metric-extension},
 proposition~\ref{thm:partial-is-smooth-sec} and
 proposition~\ref{thm:covariant-derivative-h0} -- which is then mapped to a
-positive-definite section of \(\Sym^2 T H^1(I, M)\) by the canonical
-isomorphism \(\coprod_\gamma H^1(\gamma^* TM) \isoto T H^1(I, M)\).
+positive-definite section of \(\Sym^2 T H^1(I, M)\) by the induced isomorphism
+\(\Sym^2 \coprod_\gamma H^1(\gamma^* TM) \isoto \Sym^2 T H^1(I, M)\).