Solution
= Solution
The <dual basis> is the collection of <linear functionals> characterized by
$$
\boxed{e'_i(e_j)=\delta_{ij}.}
$$
Explicitly, $e'_i(\sum_j\alpha_je_j)=\alpha_i$. Every <linear functional> $\ell\in V'$ satisfies $\ell=\sum_i\ell(e_i)e'_i$, proving that the displayed functionals span the <dual space>. Evaluating a zero <linear combination> at each $e_j$ makes all its coefficients zero, so they are also <linearly independent>. Thus they form a <basis> of $V'$.