Let $(x^*_i)_{i\in I}$ be a family of linear functional on a vector space $E$ over a field $K$ and $(\eta_i)_{i\in I}$ a family of elements of $K$. Let $F'$ be the span of $(x^*_i)_{i\in I}$ in $E^*$.
Suppose for all $\rho\in K^{(I)}_d$ such that $\sum_{i\in I}x^*_i\rho_i=0$ we have $\sum_{i\in I}\eta_i\rho_i=0$. The author asserts that this condition implies that there exists a linear mapping $f:F'\rightarrow K$ such that $f(x^*_i)=\eta_i$ for all $i$. I don't see what justifies the existence of such a linear map. Explanation please?
Edit:
There exists some $J\subset I$ such that $(x^*_i)_{i\in J}$ is a basis of $F'$. Then there exists a linear mapping $f:F'\rightarrow K$ s.t. $f(x^*_i)=\eta_i$ for all $i\in J$. Is there way to extend the equalities to the all of $I$?