If $x$ is not belongs to span of $y$ find a functional on X with $\phi x=1$ and $\phi y=0$? If $x$ is not belongs to span of $y$ find a linear functional on X with $\phi x=1$ and $\phi y=0$ ? where $\phi: X->F$ ,F is a field ,X is in Banach space 

Suppose $\;y\neq0\;$, then observe that $\;\{x,y\}\;$ is a linearly independent set. Complete it to a basis $\;\{x,y,x_i\}_{i\in I}\;$ of $\;X_F\;$ (you may need AC to do this. I am not sure...), and define
$$\phi x=1\;,\;\;\phi y=\phi x_i=0$$
and extend the definition by linearity.
If $\;y=0\;$ then, as $\;x\neq0\;$ , extend $\;\{x\}\;$ to a basis of $\;X_F\;$ as before and define in a similar way.
