The context is the following :
Proposition : Let $E$ be a real Banach space, $\Omega \subseteq E$ open, $\omega \in C^{1}(\Omega,E^{*})$ a closed 1-form. Let be $\gamma_{0},\gamma_{1}$ $C^{2}$ two $\gamma$-homotopic paths with fixed endpoints. It holds that $\int_{\gamma_{0}} \omega = \int_{\gamma_{1}} \omega$.
I do understand the proof I have of the above statement, which goes on defining $\phi(s) = \int_{\gamma(s,\bullet)} \omega$ and prooving that it's constant;
What I don't get is the assumption (without loss of generality) that $\gamma$, the homotopy between $\gamma_{0},\gamma_{1}$, a priori $C^{0}$, can actually be considered $C^{2}$.
Are there any known approximation Thorems for this? I thought about using Stone-Weierstrass or trying to approximate the homotopy with piecewise $C^{2}$ polygonals taken in appropriate dense set, failing.
I found some references of what I'm looking for only between manifolds and smooth maps in the following links, but since manifold are above my knowledge i didn't find find any solution.
Any help or comment would be appreciated,thanks.