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

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -34,13 +34,14 @@ maximizing functions by studying the critical points of \(f\). More generally,
 modern calculus of variations is concerned with the study of critical points of
 smooth functionals \(\Gamma(E) \to \mathbb{R}\), where \(E \to M\) is a smooth
 fiber bundle over a finite-dimensional manifold \(M\) and \(\Gamma\) is a given
-section functor, such as smooth sections or Sobolev sections -- notice that by
-taking \(E = M \times N\) the manifold \(\Gamma(E)\) is naturally identified
-with a space of functions \(M \to N\), which gets us back to the original case.
+section functor, such as smooth sections, continuous sections or Sobolev
+sections -- notice that by taking \(E = M \times N\) the manifold \(\Gamma(E)\)
+is naturally identified with a space of functions \(M \to N\), which gets us
+back to the original case.
 
 In these notes we hope to provide a very brief introduction to modern theory
 the calculus of variations by exploring one of the simplest concrete examples
-of the previously described program: we study the differential structure of the
+of the previously described program. We study the differential structure of the
 Banach manifold \(H^1(I, M)\) of class \(H^1\) curves in a finite-dimensional
 Riemannian manifold \(M\), which encodes the solution to the \emph{classic}
 variational problem: that of geodesics. Hence the particular action functional