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Show that if a normed space $X $ has a linearly independent subset of $n$ elements, so does the dual space $X'$

My attempt :

$\text{Given that a normed space $X$ has a linearly indepenedent susbset of $n-$ element}\tag1$

let the subset be $S=\{ e_1,e_2,e_3,....,e_n\}$

Define $e_i \in X$ by $f_j(e_i)= \delta_{ij} = \begin{cases} 1 & i=j \\0 , & i \neq j \end{cases}$ where $1\le i\le n$ and $1\le j\le n$

From $(1)$ we have $c_1e_1+...+c_ne_n=0\implies c_1f(e_1)+...+c_nf(e_n)=0$

After that im not able to proceed further