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

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -511,7 +511,7 @@ however, is the following consequence of Morse's theorem.
 
   Let \(\gamma\) be as in \textbf{(i)}. Since \(\gamma\) has no conjugate
   points, it follows from Morses's index theorem that \(T_\gamma^- \Omega_{p q}
-  M = 0\). Furtheremore, by noting that any piece-wise smooth \(X \in \ker
+  M = 0\). Furthermore, by noting that any piece-wise smooth \(X \in \ker
   A_\gamma\) is Jacobi field vanishing at \(p\) and \(q\) one can also show
   \(T_\gamma^0 \Omega_{p q} M = 0\). Hence \(T_\gamma \Omega_{p q} M =
   T_\gamma^+ \Omega_{p q} M\) and therefore \(d^2 E\!\restriction_{\Omega_{p q}
@@ -528,7 +528,7 @@ however, is the following consequence of Morse's theorem.
     v & \mapsto \exp_\gamma(\delta (v_1 \cdot X_1 + \cdots + v_k \cdot X_k))
   \end{align*}
 
-  Clearly \(i\) is an immersion for small enought \(\delta\). Moreover, from
+  Clearly \(i\) is an immersion for small enough \(\delta\). Moreover, from
   (\ref{eq:energy-taylor-series}) and
   \[
     E(i(v))
@@ -546,10 +546,10 @@ showing that the length functional \(\ell : H^1(I, M) \to \RR\) is smooth and
 that its Hessian \(d^2 \ell_\gamma\) is given by \(C \cdot d^2 E_\gamma\) for
 some \(C > 0\).
 
-Secondly, inlike the classical formulation we only consider curves in an
+Secondly, unlike the classical formulation we only consider curves in an
 \(H^1\)-neighborhood of \(\gamma\) -- instead of a neighborhood of \(\gamma\)
 in \(\Omega_{p q} M\) in the uniform topology. On the other hand, part
-\textbf{(ii)} is definitevely an improvement of the classical formulation: we
+\textbf{(ii)} is definitively an improvement of the classical formulation: we
 can find curves \(\eta = i(v)\) shorter than \(\gamma\) already in an
 \(H^1\)-neighborhood of \(\gamma\).