# Thread: Infinite Dimensional Vector Spaces and Linear Operators

1. ## Infinite Dimensional Vector Spaces and Linear Operators

Give a example where V is an infinite dimensional vector space , $T_1, T_2$are linear operators on V and $T_1T_2$ (a composition) is a bijection but T_1 and T_2 are not bijections

2. Originally Posted by I-Think
Give a example where V is an infinite dimensional vector space , $T_1, T_2$are linear operators on V and $T_1T_2$ (a composition) is a bijection but T_1 and T_2 are not bijections
Think of the space $\mathbb{R}[x]$ and don't think too hard about the mappings...do what's natural.

3. What is the space R[x]?

4. Originally Posted by I-Think
What is the space R[x]?

The real vector space whose elements are the polynomials with real coefficients on the unknown $x$ considering the standard operation sum an product by a real number.

5. Originally Posted by I-Think
Give a example where V is an infinite dimensional vector space , $T_1, T_2$are linear operators on V and $T_1T_2$ (a composition) is a bijection but T_1 and T_2 are not bijections

Take $V:=\{\,\{x_n\}_{n=1}^\infty\;;\;x_n\in\mathbb{R}\, \,\forall n\in\mathbb{N}\} =$ the vector space of all

real sequences with coordinatewise addition and multiplication by scalar, and take

$T_2(\{x_n\}):=\{0,x_1,x_2,\ldots\}\,,\,\,T_1(\{x_n \}):=\{x_2,x_3,\ldots\}$ , and prove these two do the trick.

Tonio