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

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -427,11 +427,7 @@ essential for stating Morse's index theorem.
 
 \begin{corollary}
   Given a critical point \(\gamma\) of \(E\!\restriction_{\Omega_{p q} M}\),
-  \(A_\gamma\) has either finitely many eigenvalues including \(1\) or
-  infinitely many eigenvalues not equal to \(1\), in which case the only
-  accumulation point of the set of eigenvalues of \(A_\gamma\) is \(1\) and
-  \(1\) is a spectral value but not an eigenvalue. In particular, there is an
-  orthogonal decomposition
+  there is an orthogonal decomposition
   \[
     T_\gamma \Omega_{p q} M
     = T_\gamma^- \Omega_{p q} M
@@ -441,9 +437,9 @@ essential for stating Morse's index theorem.
   where \(T_\gamma^- \Omega_{p q} M\) is the finite-dimensional subspace
   spanned by eigenvectors with negative eigenvalues, \(T_\gamma^0 \Omega_{p q}
   M = \ker A_\gamma\) and \(T_\gamma^+ \Omega_{p q} M\) is the proper Hilbert
-  subspace spanned by eigenvectors with positive eigenvalues. The same holds
-  for critical points \(\gamma\) of \(E\!\restriction_{\Lambda M}\) and
-  \(T_\gamma \Lambda M\).
+  subspace given by the closure of the subspace spanned by eigenvectors with
+  positive eigenvalues. The same holds for critical points \(\gamma\) of
+  \(E\!\restriction_{\Lambda M}\) and \(T_\gamma \Lambda M\).
 \end{corollary}
 
 \begin{definition}\label{def:morse-index}