Let $X$ be an infinite-dimensional vector space, and let $x_0$ be an element of $X$ such that $f(x_0)=0$ for every linear functional $f$ defined on $X$. Then can we prove that $x_0$ is the zero vector in $X$? Or, is it possible to find a non-zero $x_0$ whose image under every linear functional is zero?
In the case of $X$ being finite-dimensional, we can prove that if $f(x_0)=0$ for every linear functional $f$ on $X$, then $x_0$ must be the zero vector. What is the situation in the infinite-dimensional case?