# Infinite Dimensional Vector Spaces and Linear Operators

• Mar 8th 2011, 10:36 PM
I-Think
Infinite 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 PM
Drexel28
Quote:

Originally Posted by I-Think
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

Think of the space $\displaystyle \mathbb{R}[x]$ and don't think too hard about the mappings...do what's natural.
• Mar 8th 2011, 11:30 PM
I-Think
What is the space R[x]?
• Mar 9th 2011, 12:10 AM
FernandoRevilla
Quote:

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 $\displaystyle x$ considering the standard operation sum an product by a real number.
• Mar 9th 2011, 02:28 AM
tonio
Quote:

Originally Posted by I-Think
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