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Math Help - Infinite Dimensional Vector Spaces and Linear Operators

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    Senior Member I-Think's Avatar
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    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
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by I-Think View Post
    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.
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    Senior Member I-Think's Avatar
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    What is the space R[x]?
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by I-Think View Post
    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.
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    Quote Originally Posted by I-Think View Post
    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
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