Geometry of Bounded Frechet Manifolds

In this paper we develop the geometry of bounded Fr\'echet manifolds. We prove that a bounded Fr\'echet tangent bundle admits a vector bundle structure. But the second order tangent bundle $T^2M$ of a bounded Fr\'echet manifold $M$, becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. As an application, we prove the existence and uniqueness of the integral curve of a vector field on $M$

Sard's theorem for mappings between Fr\'echet manifolds

In this paper we prove an infinite-dimensional version of Sard's theorem for Fr\'{e}chet manifolds. Let $M$ and $N$ be bounded Fr\'{e}chet manifolds such that the topologies of their model Fr\'{e}chet spaces are defined by metrics with absolutely convex balls. Let $f: M \rightarrow N$ be an $MC^k$-Lipschitz-Fredholm map with k > \max \lbrace {\Ind f,0} \rbrace . Then the set of regular values of $f$ is residual in $N$