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

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -33,16 +33,16 @@ Finally, we may define\dots
   From now on we fix \(I = [0, 1]\).
 \end{note}
 
-We should note that every peace-wise smooth curve \(\gamma : I \to M\) is a
+We should note that every piece-wise smooth curve \(\gamma : I \to M\) is a
 class \(H^1\) curve. This answer raises and additional question though: why
 class \(H^1\) curves? The classical theory of the calculus of variations -- as
 described in \cite[ch.~5]{gorodski} for instance -- is usually exclusively
-concerned with the study of peace-wise smooth curves, so the fact that we are
+concerned with the study of piece-wise smooth curves, so the fact that we are
 now interested a larger class of curves, highly non-smooth curves in fact,
 \emph{should} come as a surprise to the reader.
 
 To answer this second question we return to the case of \(M = \RR^n\). Denote
-by \({C'}^\infty(I, \RR^n)\) the space of peace-wise curves in \(\RR^n\). As
+by \({C'}^\infty(I, \RR^n)\) the space of piece-wise curves in \(\RR^n\). As
 described in section~\ref{sec:introduction}, we would like \({C'}^\infty(I,
 \RR^n)\) to be a Banach manifold under which both the energy functional and the
 length functional are smooth maps. As most function spaces, \({C'}^\infty(I,
@@ -141,7 +141,7 @@ find\dots
 \begin{proposition}
   Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
   Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections
-  of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise
+  of \(E\) is the completion of the space \({C'}^\infty(E)\) of piece-wise
   smooth sections of \(E\) under the inner product given by
   \[
     \langle \xi, \eta \rangle_1
@@ -199,7 +199,7 @@ find\dots
 \end{note}
 
 We are particularly interested in the case of the pullback bundle \(E =
-\gamma^* TM \to I\), where \(\gamma : I \to M\) is a peace-wise smooth curve.
+\gamma^* TM \to I\), where \(\gamma : I \to M\) is a piece-wise smooth curve.
 \begin{center}
   \begin{tikzcd}
     \gamma^* TM \arrow{r} \arrow[swap]{d}{\pi} & TM \arrow{d}{\pi} \\