Exercise 2.6.12

Let $\beta =\{x_1,x_2,\dots ,x_n\}$ be a basis for $V$. Then we know the functional $\widehat{x_i}\in V^{\text{∗∗}}$ means $\widehat{x_i}(f)=f(x_i)$ for all funtional $f$ in $V^{\text{∗}}$. On the other hand, we have the dual basis ${\beta }^{\text{∗}}=\{f_1,f_2,\dots ,f_n\}$ is defined by $f_i(x_j)={\delta }_{\mathit{ij}}$ for all $i=1,2,\dots ,n$ and $j=1,2,\dots ,n$ such that $f_i$ is lineaer. And we can further ferret what elements are in ${\beta }^{\text{∗∗}}$. By definition of ${\beta }^{\text{∗∗}}$ we know ${\beta }^{\text{∗∗}}=\{F_1,F_2,\dots ,F_n\}$ and $F_i(f_j)={\delta }_{\mathit{ij}}$ and $F_i$ is linear. So we may check that whether $F_i=\widehat{x_i}$ by

Since they are all linear functional and the value of them meets at basis $\beta$, they are actually equal by the Corollary after Theorem 2.6.