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

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -124,6 +124,9 @@ Banach spaces. We begin by the former.
   class \(C^n\) for all \(n > 0\).
 \end{definition}
 
+The following lemma is also of huge importance, and it is known as \emph{the
+chain rule}.
+
 \begin{lemma}\label{thm:chain-rule}
   Given Banach spaces \(V_1\), \(V_2\) and \(V_3\), open subsets \(U_1 \subset
   V_1\) and \(U_2 \subset V_2\) and two smooth maps \(f : U_1 \to U_2\) and \(g
@@ -166,19 +169,52 @@ subject can be found in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
   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
-  each \(i\) with \(p \in U_i\): each chart \(\varphi_i : U_i \to M\) induces a
-  norm in \(T_p M\)\footnote{In general $T_p M$ is not a normed space, since
-  the norms induced by two distinct choices of chard need not to coincide.
-  However, since all $V_i$'s are isomorphic as Banach spaces this norms are
-  all mutually equivalent, so that they define a (unique) topology in $T_p M$.}
-  given by the pullback of the norm of \(V_i\) through the linear isomorphism
+\end{definition}
+
+\begin{definition}
+  Given \(p \in M\) and a chart \(\varphi_i : U_i \to V_i\) with \(p \in U_i\),
+  let
   \begin{align*}
-    \phi_i :    T_p M & \isoto  V_i                          \\
-             [\gamma] & \mapsto (\varphi_i \circ \gamma)'(0)
+    \phi_{p, i} :    T_p M & \to     V_i                          \\
+                  [\gamma] & \mapsto (\varphi_i \circ \gamma)'(0)
   \end{align*}
 \end{definition}
 
+\begin{lemma}
+  Given \(p \in M\) and a chart \(\varphi_i\) with \(p \in U_i\), \(\phi_{p,
+  i}\) is a linear isomorphism. For any given charts \(\varphi_i, \varphi_j\),
+  the pullback of the norms of \(V_i\) and \(V_j\) by \(\varphi_i\) and
+  \(\varphi_j\) respectively define equivalent norms in \(T_p M\). In
+  particular, any choice chart gives \(T_p M\) the structure of a topological
+  vector space, and this topology is independent of this choice\footnote{In
+  general $T_p M$ is not a normed space, since the norms induced by two
+  distinct choices of chard need not to coincide. Nevertheless, the topology
+  induced by this norms is the same.}.
+\end{lemma}
+
+\begin{proof}
+  The first statement about \(\phi_{p, i}\) being a linear isomorphism should
+  be clear from the definition of \(T_p M\). The second statement about the
+  equivalence of the norms is equivalent to checking that \(\phi_{p, i} \circ
+  \phi_{p, j}^{-1} : V_j \to V_i\) is continuous for each \(i\) and \(j\) with
+  \(p \in U_i\) and \(p \in U_j\).
+
+  But this follows immediately from the identity
+  \[
+    \begin{split}
+      (\phi_{p, i} \circ \phi_{p, j}^{-1}) v
+      & = (\varphi_i \circ \varphi_j^{-1} \circ \gamma_v)'(0) \\
+      \text{(chain rule)}
+      & = (d \varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} \dot\gamma_v(0) \\
+      & = (d \varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} v
+    \end{split}
+  \]
+  where \(v \in V_j\) and \(\gamma_v : (-\epsilon, \epsilon) \to V_j\) is any
+  smooth curve with \(\gamma_v(0) = \varphi_j(p)\) and \(\dot\gamma_v(0) = v\).
+  In other words, \(\phi_{p, i} \circ \phi_{p, j}^{-1} = (d \varphi_i \circ
+  \varphi_j^{-1})_{\varphi_j(p)}\) is continuous by definition.
+\end{proof}
+
 Notice that a single Banach manifold may be ``modeled after'' multiple Banach
 spaces, in the sense that the \(V_i\)'s of definition~\ref{def:banach-manifold}
 may vary with \(i\). Lemma~\ref{thm:chain-rule} implies that for each \(i\) and