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Let $K$ be a field. Given $V$ a $K$ vector-space containing all sequences $a=(a_n)_{n\in \mathbb N}$ with values from $K$, such that $a_n\neq 0$ for only finitely many $n\in \mathbb N$.

I want to to find $f,g\in\mathrm{End}(V)$ such that $f\circ g=id_V$ and $f,g\not\in\mathrm{End}(V)^{\times}.$

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You may use "conveyor belt" maps, for instance $f(a_{0},a_{1},\dotsc)=(a_{1},a_{2},\dotsc)$ and $g(a_{0},a_{1},\dotsc)=(0,a_{0},a_{1},a_{2},\dotsc)$.

Analogously, in a polynom or regular functions space, you may use derivation and primitivation relatively to a given point. That's essentially what I've done.

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    The "conveyor belt" is usually called _left shift operator_ for obvious reason. (It is always nice to know names to remember and to search.) This is an example that retraction and section are not necessary isomorphism. – 2017-02-24