1

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 $f,g\not\in\mathrm{End}(V)^{\times}.$

user26857
  • 52,094
Marc
  • 2,074

1 Answers1

4

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.

Tig la Pomme
  • 641
  • 2
    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. – Orat Feb 24 '17 at 12:49