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

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -210,8 +210,8 @@ subject can be found in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
     \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
+  smooth curve with \(\gamma_v(0) = \varphi_j(p)\) and \(\dot\gamma_v(0) = v\):
+  \(\phi_{p, i} \circ \phi_{p, j}^{-1} = (d \varphi_i \circ
   \varphi_j^{-1})_{\varphi_j(p)}\) is continuous by definition.
 \end{proof}
 

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -92,7 +92,7 @@ topology. This hints at the fact that in order to \(E\) and \(\ell\) to be
 continuous maps we need to control both \(\gamma\) and \(\dot\gamma\). Hence a
 natural candidate for a norm in \({C'}^\infty(I, \RR^n)\) is 
 \[
-  \norm{\gamma}_1 = \norm{\gamma}_2 + \norm{\dot\gamma}_2,
+  \norm{\gamma}_1^2 = \norm{\gamma}_2^2 + \norm{\dot\gamma}_2^2,
 \]
 which is, of course, the norm induced by the inner product \(\langle \, ,
 \rangle_1\) -- here \(\norm{\cdot}_2\) denote the norm of \(L^2(I, \RR^n)\).