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

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -47,7 +47,7 @@ differential structure of \(H^1(I, M)\) and its canonical Riemannian metric. In
 section~\ref{sec:aplications} we sutdy the critical points of the energy
 functional \(E : H^1(I, M) \to \RR\) and describe several applications to the
 study of the geodesics of \(M\). Bofore moving to the next section, however, we
-would like to review the basics of the theory of real Banach manifolds. 
+would like to review the basics of the theory of real Banach manifolds.
 
 \subsection{Banach Manifolds}
 
@@ -59,7 +59,7 @@ translated to the realm of Banach manifolds\footnote{The real difficulties with
 Banach manifolds only show up while proving certain results, and are maily due
 to complications regarding the fact that not all closed subspaces of a Banach
 space have a closed complement}. The reason behind this is simple: it turns out
-that calculus has nothing to do with \(\RR^n\). 
+that calculus has nothing to do with \(\RR^n\).
 
 What we mean by this last statement is that none of the fundamental ingrediants
 of calculus -- the ones necessary to define differentiable functions in
@@ -94,7 +94,7 @@ Banach spaces. We begin by the former.
   1\): a function \(f : U \to W\) of class \(C^n\) is called
   \emph{differentiable of class \(C^{n + 1}\)} if the map
   \[
-    d^n f : 
+    d^n f :
     U \to \mathcal{L}(V, \mathcal{L}(V, \cdots \mathcal{L}(V, W)))
     \cong \mathcal{L}(V^{\otimes n}, W)
     \footnote{Here we consider the \emph{projective tensor product} of Banach
@@ -144,7 +144,7 @@ subject can be found in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
   and \(p \in M\), the tangent space \(T_p M\) of \(M\) at \(p\) is the
   quotient of the space \(\{ \gamma : (- \epsilon, \epsilon) \to M \mid \gamma
   \ \text{is smooth}, \gamma(0) = p \}\) by the equivalence relation that
-  identifies two curves \(\gamma\) and \(\eta\) such that 
+  identifies two curves \(\gamma\) and \(\eta\) such that
   \((\varphi_i \circ \gamma)'(0) = (\varphi_i \circ \eta)'(0)\) for all \(i\)
   with \(p \in U_i\).
   The space \(T_p M\) is a topological vector space isomorphic to \(V_i\) for