Let $W$ be a (real) Banach space and $M$ be a (finite-dimensional) Manifold. Consider the trivial $W$-bundle $\pi: E \to M$ over $M$ with $E := W \times M$ and $\pi$ the second-factor projection. Denote as usual by $TM$ the tangent bundle of $M$ and by by $\Gamma(E)$ and $\Gamma(TM)$ the space of smooth sections of $E$, resp. $TM$. My question is the following:
Does there exists a canonical connection $\nabla$ on $E$ ?
By connection, I mean an $\mathbb R$-bilinear map $\nabla: \Gamma(TM) \times \Gamma(E) \to \Gamma(E)$ that is also $C^\infty(M,\mathbb R)$-linear in the first factor and satisfies the Leibnitz-identity \begin{equation} \nabla_X(f \omega) = X(f)\omega + f \nabla_X(\omega) \end{equation} for all $X \in \Gamma(TM)$, all $f \in C^\infty(M,\mathbb R)$ and all $\omega \in \Gamma(E)$.
First, it is easily seen that $\Gamma(E) = C^\infty(M,W)$, the space of (Frechét)-smooth functions from $M$ with values in $W$. Therefore, if $W$ is finite-dimensional, one can construct a canonical connection as follows:
Choosing a basis $(b_i)_{i=1}^N$ on $W$ with $N := \dim_{\mathbb R}(W)$, one can see that any $\omega \in \Gamma(E)$ can be written uniquely as $\omega = \sum_{i=1}^N \omega_ib_i$ with $\omega_i \in C^\infty(M,\mathbb R)$. Then one defines $\nabla$ as a above by $\nabla_X(\omega) := \sum_{i=1}^N X(\omega_i)b_i$. Using elementary rules of differentiation, one can verify that $\nabla$ is canonical, in the sense that is doesn't depend on the particular choice of basis $(b_i)_{i=1}^N$.
It is not obvious to me how to extend the argument of the previous paragraph to the case when $W$ is infinite-dimensional. Any help is appreciated.
The answer to your question is yes. In fact, every trivialization of a vector bundle (or any fiber bundle, actually) yields a trivial connection. This is exactly the connection you describe in the finite-rank case, but there is another way to describe it which does not involve a frame.
As you have noted yourself, a section of $E$ is nothing more than a function $M\to W$. Hence, for a section $s$ and a tangent vector $X$, we can define $$\nabla_Xs=ds(X),$$where $ds$ is the usual differential of a smooth function. It should take a few minutes to see that in the finite-rank case this is the connection described in your post.