TITLE: If $x, y \in X$ with $x \neq y$, then there exists $f \in X^*$ such that $f(x) \neq f(y)$.
QUESTION [8 upvotes]: Let $X$ be a normed linear space. Prove that if $x, y \in X$ with $x \neq y$, then there exists $f \in X^*$ such that $f(x) \neq f(y)$.
Here $X^*$ denotes the dual space of $X$.
I am getting some smell of using Hahn Banach theorem but not able to prove it. Need some hints.

REPLY [2 votes]: The Hann-Banach everyone is talking about but not spelling out:
Take $$H:=span \{x-y\}$$ and define a linear functional on $H$ as $$f(t(x-y)):=t \ $$ An extension of this $f$ to all of your space will work.