diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -3,10 +3,10 @@
Known as \emph{global analysis}, or sometimes \emph{non-linear functional
analysis}, the field of study dedicated to the understanding of
infinite-dimensional manifolds has seen remarkable progress in the past several
-decades. Among its numerous contributions to geometry at large, perhaps the
-greatest achievement in global analysis in the last century was the recognition
-of the fact that many interesting function spaces possesses natural
-differentiable structures -- which are usually infinite-dimensional.
+decades. Among numerous discoveries, perhaps the greatest achievement in global
+analysis in the last century was the recognition of the fact that many
+interesting function spaces possess natural differentiable structures -- which
+are usually infinite-dimensional.
As it turns out, many local problems regarding maps between finite-dimensional
manifolds can be translated to global questions about the geometry of function
@@ -36,7 +36,7 @@ smooth functionals \(\Gamma(E) \to \RR\), where \(E \to M\) is a smooth fiber
bundle over a finite-dimensional manifold \(M\) and \(\Gamma\) is a given
section functor, such as smooth sections or Sobolev sections -- notice that by
taking \(E = M \times N\) the manifold \(\Gamma(E)\) is naturally identified
-with a space of functions \(M \to N\), getting back at the original case.
+with a space of functions \(M \to N\), which gets us back to the original case.
In this notes we hope to provide a very brief introduction to modern theory the
calculus of variations by exploring one of the simplest concrete examples of
@@ -82,11 +82,11 @@ While it is certainly true that Banach spaces can look alien to someone who has
never ventured outside of the realms of Euclidean space, Banach manifolds are
surprisingly similar to their finite-dimensional counterparts. As we'll soon
see, most of the usual tools of differential geometry can be quite easily
-translated to the realm of Banach manifolds\footnote{The real difficulties with
-Banach manifolds only show up while proving certain results, and are mainly 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 \(\RR^n\).
+translated to the context of Banach manifolds\footnote{The real difficulties
+with Banach manifolds only show up while proving certain results, and are
+mainly 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 \(\RR^n\).
What we mean by this last statement is that none of the fundamental ingredients
of calculus -- the ones necessary to define differentiable functions in
@@ -104,24 +104,24 @@ Banach spaces. We begin by the former.
\[
\frac{\norm{f(p + h) - f(p) - d f_p h}}{\norm{h}} \to 0
\]
- as \(h \to 0\) in \(U\).
+ as \(h \to 0\) in \(V\).
\end{definition}
\begin{definition}
Given Banach spaces \(V\) and \(W\) and an open subset \(U \subset V\), a
continuous map \(f : U \to W\) is called \emph{differentiable of class
\(C^1\)} if \(f\) is differentiable at \(p\) for all \(p \in U\) and the
- derivative map
+ \emph{derivative map}
\begin{align*}
df: U & \to \mathcal{L}(V, W) \\
p & \mapsto d f_p
\end{align*}
is continuous. Since \(\mathcal{L}(V, W)\) is a Banach space under the
operator norm, we may recursively define functions of class \(C^n\) for \(n >
- 1\): a function \(f : U \to W\) of class \(C^n\) is called
- \emph{differentiable of class \(C^{n + 1}\)} if the map
+ 1\): a function \(f : U \to W\) of class \(C^{n - 1}\) is called
+ \emph{differentiable of class \(C^n\)} if the map
\[
- d^n f :
+ d^{n - 1} 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
@@ -147,13 +147,13 @@ chain rule}.
As promised, this simple definitions allows us to expand the usual tools of
differential geometry to the infinite-dimensional setting. In fact, in most
-cases it suffices to straight up copy the definition of the finite-dimensional
-case. For instance, as in the finite-dimensional case we may call a map \(M \to
-N\) between Banach manifolds \(M\) and \(N\) \emph{smooth} if it can be locally
-expressed as a smooth function between open subsets of the model spaces. As
-such, we will only provide the most important definitions: those of a Banach
-manifold and its tangent space at a given point. Complete accounts of the
-subject can be found in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
+cases it suffices to simply copy the definition of the finite-dimensional case.
+For instance, as in the finite-dimensional case we may call a map between
+Banach manifolds \(M\) and \(N\) \emph{smooth} if it can be locally expressed
+as a smooth function between open subsets of the model spaces. As such, we will
+only provide the most important definitions: those of a Banach manifold and its
+tangent space at a given point. Complete accounts of the subject can be found
+in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
\begin{definition}\label{def:banach-manifold}
A Banach manifold \(M\) is a Hausdorff topological space endowed with a