If two elements are different there is a functional under where the image is different

Question

I have the following exercise in functional analysis: Let E be a normed space and $x,y$ different vectors. Prove or disprove finding a counterexample that it exist a function $f \in E^*$ such that $f(x) \not= f(y)$. I tried do define a function on $<x,y> \subset E $ with different images for $x,y$ and than to extend this with hahn Banach to $E$, but I dont know how to define this. Is this statement even true or is there a counter example?

Answer 1

Yes, the statement is true, and your approach is kind of the way to go. But it is better to state the problem in the following equivalent and more useful way:

Given $x\in E$ there exists $f\in E^*$ such that $f(x)\ne0$.