I'm trying to prove that the following statement is wrong:
Let $V$ be a vector space. For all $x \in V$ there exists $f \in V^* s.t. f(x) \neq 0$
Counterexample:
Since $V$ is a vector space: $0 \in V$. And $f \in V^*$ is a linear application, since $V^*$ is the dual space. And we know that for every linear application $f(0)=0$. So the statement is wrong.
Am I right in my conclusion?