# Thread: Normed space and dual space

1. ## Normed 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}

2. Originally Posted by geezeela
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}

$\displaystyle \exists 0\neq x\in X\Longrightarrow$ 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

3. Originally Posted by geezeela
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}
More high-powered (assuming you can use this theorem--which you probably can't) you can use the fact that in general $\displaystyle \dim V\leqslant\dim\text{Hom}(V,F)$.

4. Originally Posted by tonio
$\displaystyle \exists 0\neq x\in X\Longrightarrow$ 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
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.

5. Originally Posted by Jose27
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.

6. Originally Posted by Drexel28
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.