Let $V$ be a infinite dimensional vector space over a field $K$, and $\{v_i\}_{i\in I}$ be a basis of $V$. For each $i\in I$, let $f_i: V\to K$ be defined by $f_i(v_j)=\delta_{ij}$. Prove that $\{f_i\}_{i\in I}$ is linearly independent but does not span the dual space $V^*$.
I can prove that $\{f_i\}_{i\in I}$ is linearly independent. But I cannot find a counterexample in $V^*$ that is not spanned by $\{f_i\}_{i\in I}$. Does anyone has any idea on how to create this counterexample? This is the difference between finite and infinite dimensional vector space, right?
Thank you very much!