Let $X$ be a finite set and let $V$ be the space of all maps from $X$ to $F$. For each $x \in X$, consider the map $l_x : V \to F$ sending $f$ to $f(x)$. Prove that the family $\{l_x :\, x \in X\}$ is a basis for $V^*$.
The idea to prove that it is the basis of the dual space is to use the Kronecker's delta, but I don't know how to start, the operator definition confuses me, can you help me?
Since $X$ is a finite set, let $x_1,x_2,\dots,x_n$ be the distinct elements of $X$. Now, for each $i$ between $1$ and $n$, consider the function $f_i \in V$ defined by $$f_i(x_j) = \delta_{ij} = \begin{cases} 1 & \textrm{if } i = j \\ 0 & \textrm{if } i \neq j \end{cases}$$
Exercise. Prove that the functions $f_1,f_2,\dots,f_n$ form a basis for $V$.
Now, in order to prove that $\{l_x\}_{x \in X} = \{l_{x_1}, l_{x_2}, \dots, l_{x_n}\}$ is a basis for $V^*$, we need to prove that only the linear independence (why?).
So, suppose that the zero $0 \in V^*$ can be written as $$0 = a_1l_{x_1} + a_2l_{x_2} + \cdots + a_nl_{x_n}$$ for some choice of scalars $a_1,a_2,\dots,a_n$ in $F$. Now, evaluating the preceding equation in $f_1$ we obtain \begin{align} 0 = 0(f_1) &= (a_1l_{x_1} + a_2l_{x_2} + \cdots + a_nl_{x_n})(f_1) \\ &= a_1l_{x_1}(f_1) + a_2l_{x_2}(f_1) + \cdots + a_nl_{x_n}(f_1) \\ &= a_1f_1(x_1) + a_2f_1(x_2) + \cdots + a_n f_1(x_n) = a_1 \end{align} and similarly, $a_2 = \cdots = a_n = 0$.