Let X be normed space and X' be dual its dual space if $\displaystyle X\neq \left\{ 0 \right\} $, show tjat X' cannot be {0}

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- Jun 5th 2011, 06:29 PMgeezeelaNormed space and dual space
Let X be normed space and X' be dual its dual space if $\displaystyle X\neq \left\{ 0 \right\} $, show tjat X' cannot be {0}

- Jun 5th 2011, 07:03 PMtonio

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

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

Then $\displaystyle 0\neq f\in X'$

Tonio - Jun 5th 2011, 07:13 PMDrexel28
- Jun 5th 2011, 10:36 PMJose27
- Jun 5th 2011, 10:38 PMDrexel28
- Jun 5th 2011, 10:46 PMJose27