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

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- Mar 8th 2011, 10:36 PMI-ThinkInfinite Dimensional Vector Spaces and Linear Operators
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

- Mar 8th 2011, 11:07 PMDrexel28
- Mar 8th 2011, 11:30 PMI-Think
What is the space R[x]?

- Mar 9th 2011, 12:10 AMFernandoRevilla
- Mar 9th 2011, 02:28 AMtonio

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