I am preparing an introduction to banach manifolds and currently I am working on exact sequences of vector bundles of banach manifolds. Are there any nice examples (i.e. easy to understand for people who do not have any knowledge on this subject yet) for a short exact sequence of vector bundle morphisms which splits?
I have thought one could try to use the fundamental theorem of calculus short exact sequence (cf. Terry Tao's comment) given by
$0 \to \mathbb{R} \to C^{\infty}(\mathbb{R}) \overset{\frac{\text{d}}{\text{d}x}}{\to}C^{\infty}(\mathbb{R}) \to 0.$
However, I am not seeing how $\mathbb{R}$ on the LHS becomes a vector bundle.
One simple example is to start with a short exact sequence of finite dimensional vector spaces, i.e. a sequence $$ 0\to U\xrightarrow{f}V\xrightarrow{g}W\to 0 $$ where $f$ is injective, $g$ is surjective, and $\operatorname{im}(f)=\ker(g)$. Since all such sequences split, the corresponding sequence of trivial vector bundles will split in the corresponding way. For instance, in $\mathsf{Vect}(M)$ we have: $$ M\times 0\to M\times U\xrightarrow{\operatorname{id_M\times f}}M\times V\xrightarrow{\operatorname{id_M\times g}}M\times W\to M\times 0 $$