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Math Help - Normed space and dual space

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    Normed space and dual space

    Let X be normed space and X' be dual its dual space if  X\neq \left\{ 0 \right\} , show tjat X' cannot be {0}
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    Quote Originally Posted by geezeela View Post
    Let X be normed space and X' be dual its dual space if  X\neq \left\{ 0 \right\} , show tjat X' cannot be {0}

    \exists 0\neq x\in X\Longrightarrow extend the lin. independent set \{x\} to a basis \{x,\{x_i\}\;;\;i\in I\} of X (you may need the axiom of choice if \dim X = \infty)

    and define f: X \longrightarrow\mathbb{F}\,,\,f(x)=1\,,\,f(x_i)=0\  ,\,\forall i\in I and extend the definition to all X by linearity { \mathbb{F}= the definition field)

    Then 0\neq f\in X'

    Tonio
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    Let X be normed space and X' be dual its dual space if  X\neq \left\{ 0 \right\} , show tjat X' cannot be {0}
    More high-powered (assuming you can use this theorem--which you probably can't) you can use the fact that in general \dim V\leqslant\dim\text{Hom}(V,F).
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    Quote Originally Posted by tonio View Post
    \exists 0\neq x\in X\Longrightarrow extend the lin. independent set \{x\} to a basis \{x,\{x_i\}\;;\;i\in I\} of X (you may need the axiom of choice if \dim X = \infty)

    and define f: X \longrightarrow\mathbb{F}\,,\,f(x)=1\,,\,f(x_i)=0\  ,\,\forall i\in I and extend the definition to all X by linearity { \mathbb{F}= the definition field)

    Then 0\neq f\in X'

    Tonio
    This isn't always continous if the basis you picked is a Hamel basis, and not all spaces have a Shauder basis.

    Hint: There is an embedding of X in its double dual.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Jose27 View Post
    This isn't always continous if the basis you picked is a Hamel basis, and not all spaces have a Shauder basis.

    Hint: There is an embedding of X in its double dual.
    Perhaps the OP mean the algebraic dual and not the continuous dual.
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  6. #6
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    Quote Originally Posted by Drexel28 View Post
    Perhaps the OP mean the algebraic dual and not the continuous dual.
    Since X is assumed normed, I'm assuming he/she means the continous dual. I guess we'll have to ask him/her to clarify.
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