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?

In all of the contexts I've seen it being used, you have
$$\delta_{ij} = \begin{cases}
1, & \text{ for } i = j \\
0,  & \text{ for } i \neq j
\end{cases}\tag{1}\label{eq1A}$$
