# Normed space and dual space

• Jun 5th 2011, 06:29 PM
geezeela
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}
• Jun 5th 2011, 07:03 PM
tonio
Quote:

Originally Posted by geezeela
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
• Jun 5th 2011, 07:13 PM
Drexel28
Quote:

Originally Posted by geezeela
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 $\dim V\leqslant\dim\text{Hom}(V,F)$.
• Jun 5th 2011, 10:36 PM
Jose27
Quote:

Originally Posted by tonio
$\exists 0\neq x\in X\Longrightarrow$ 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.
• Jun 5th 2011, 10:38 PM
Drexel28
Quote:

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.
• Jun 5th 2011, 10:46 PM
Jose27
Quote:

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.