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