Given a $g$ an odd function the question is to exhibit a continuous linear functional from $C[-1,1]$ $\phi$ such that
$|\phi|=1$ , $\phi(g)=|g|_\infty$ and $\phi$ is zero on the even functions.
Given a $g$ an odd function the question is to exhibit a continuous linear functional from $C[-1,1]$ $\phi$ such that
$|\phi|=1$ , $\phi(g)=|g|_\infty$ and $\phi$ is zero on the even functions.
Let $x_0$ be a point where $g(x_0)=\Vert g\Vert_\infty$ (as $g$ is an odd function, such a point exists). Define $\phi(f)={1\over2}\bigl(f(x_0)-f(-x_0)\bigr)$.
Then $\phi$ linear. The norm of $\phi$ is 1 by virtue of the fact that $\phi(g)=\Vert g\Vert_\infty$ and $|\phi(f)|\le {1\over 2}(|f(x_0)|+|f(-x_0)|)\le\Vert f\Vert_\infty$. Also, if $f$ is even, $\phi(f)=0$.