Let $X$ be a separable, infinite dimensional Banach space. Does $X^\star$ (the set of bounded complex linear functionals) separate the points of $X$? (meaning, for every two vectors $x,y\in X$ there is some $\phi \in X^\star$ such that $\phi(x)\neq\phi(y)$). What if $X$ is not a Banach space and is just a Fréchet space?
Yes, the dual $X^*$ of every banach space $X$ separates the points of $X$. This is an immediate consequence of the Hahn-Banch theorem. A proof can be found in every introductory course on fuctional analysis. Moreover, the Hahn-Banach theorem holds in locally convex spaces. Thus statement is also true for Fréchet spaces.