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