This question might seem elementary but I cannot answer it.

Let M be an infinite dimensional vector space and $f_1, f_2, \cdots , f_r \in M^*$ be a set of linear independent vectors with $r \geq 2$. Here $M^*:=Hom_K(M, K)$ is the linear dual of $M$.

Do there exist $m \in M$ such that $f_1(m)\neq 0$ but $f_i(m)=0$ for any $i\geq 2$?

Of course the statement is obviously true for finite dimensional vector spaces $M$.

I thought about completing $\{f_1, f_2, \cdots, f_r\}$ to a basis of $M^*$ and then taking the dual basis with respect to it. But this new basis lies in $M^{**}$ which is larger then $M$.