Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth functions from $J$ into $X$, with the usual topology of uniform convergence of functions and all their derivatives on compact subsets of $J$. In particular, $M$ is a topological module over the Fr\'{e}chet algebra $C^\infty(J)$ of smooth complex-valued functions on $J$, where the module action in pointwise multiplication. I prefer to think of $M$ as sections of the trivial vector bundle over $J$ with fiber $X$ at each $t \in J$. As such, I will use the notation $M_t$ to denote this fiber, which is isomorphic to $X$.
By a connection on $M$, I mean a continuous complex linear map $\nabla: M \to M$ such that $$ \nabla(f\cdot m) = f'\cdot m + f\cdot \nabla(m), \qquad \forall f\in C^\infty(J), m \in M. $$ I will say the connection is integrable if for every $t \in J$ and every $m_t \in M_t$, there is a unique $m \in M$ with $\nabla m = 0$ and $m(t) = m_t$. I will say an integrable connection is topologically integrable if the association $m_t \mapsto m$ is a continuous linear map $\iota_t: M_t \to M$. Now one can show that every connection is of the form $\nabla = \frac{d}{dt} + F$ where $F: M \to M$ is a continuous $C^\infty(J)$-linear map, so we are talking about solving ODEs in the locally convex space $X$. If a connection is integrable, then one can define parallel transport maps $P:M_t \to M_s$ for any $t,s \in J$, which are necessarily linear isomorphisms. If the connection is topologically integrable, then $P$ is continuous, hence an isomorphism of topological vector spaces. For a topologically integrable connection, there is a continuous $C^\infty(J)$-linear map for each $t\in J$ $$ G_t: C^\infty(J, M_t) \to M $$ given by $G_t(f \otimes m_t) = f\cdot \iota_t(m_t),$ where we used the isomorphism $C^\infty(J,M_t) \cong C^\infty(J) \widehat{\otimes}_\pi M_t$ ($\widehat{\otimes}_\pi$ is the completed projective tensor product.) This map $G_t$ is like a trivialization. I have a few questions.
Is every integrable connection topologically integrable? Specifically, since I insisted $\nabla$ is continuous, does this mean that integrable implies topologically integrable?
If $\nabla$ is topologically integrable, is the map $G_t$ an isomorphism of topological $C^\infty(J)$-modules?
A few closing remarks: If $X$ is a Banach space, every connection is topologically integrable by the theory of ODEs in Banach space. I have also been able to show that topological integrability implies the trivialization is an isomorphism when $X$ is a Fr\'{e}chet space in the case where $[\frac{d}{dt}, F] = 0$, but I couldn't do it without avoiding the use of the Banach-Steinhaus theorem.
Thanks for reading.
