Suppose $X$ is a normed linear space. $\{f_i\}\subset X^*$ are linearly independent. Give a example to show that there not necessarily exist $\{e_j\}$ such that $f_i(e_j)=\delta_{ij}, i,j\geq 1$.
I understand that $X$ may not equal the space spanned by $e_1, e_2, \cdots$ and there are already many posts on MSE discussing about dual basis of infinite dimensional space. But I have not founded the answer of my question. Appreciate any help or hint!
Consider $C[0,1]$ with the sup norm. Arrange the rational numbers in $[0,1]$ in a sequence $(r_n)$. Let $f_i(x)=x(r_i)$. Each $f_i$ is a continuous linear functional on $C[0,1]$. Suppose we have $(x_j)$ such that $f_i(x_j)=\delta_{ij}$. Then $x_j(r_i)=\delta_{ij}$. So $x_j(r)=0$ for all rational numbers $r$ except one. By continuity it follows that $x_j\equiv 0$, a contradiction.