Give a example where V is an infinite dimensional vector space , $\displaystyle T_1, T_2 $are linear operators on V and $\displaystyle T_1T_2$ (a composition) is a bijection but T_1 and T_2 are not bijections
Give a example where V is an infinite dimensional vector space , $\displaystyle T_1, T_2 $are linear operators on V and $\displaystyle T_1T_2$ (a composition) is a bijection but T_1 and T_2 are not bijections
Take $\displaystyle V:=\{\,\{x_n\}_{n=1}^\infty\;;\;x_n\in\mathbb{R}\, \,\forall n\in\mathbb{N}\} =$ the vector space of all
real sequences with coordinatewise addition and multiplication by scalar, and take
$\displaystyle T_2(\{x_n\}):=\{0,x_1,x_2,\ldots\}\,,\,\,T_1(\{x_n \}):=\{x_2,x_3,\ldots\}$ , and prove these two do the trick.
Tonio