About the theorem of dual basis Theorem for dual basis: Let V be a finite dimensional vector space and $\beta={u_1,u_2,...,u_n}$ be an ordered basis for V. Then there exists a basis $\beta^*={f_1,...,f_n}$ of $V^*$ such that $f_i(u_j)=\delta_{ij}$
So what is $f_i(u_j)=\delta_{ij}$ in terms of significance in a linear functional, is that a coordinate function similar to a coordinate vector in a vector space?

The $f_i$ are linear functionals defined on $V$, with values in $\Bbb F$.  The Kronecker Delta is defined by $\delta_{ij}=\begin{cases}1,i=j\\0,i\ne j\end{cases}$
