Non-Traditional Definition of Dual Basis in $\mathbb{C}^N$ From an undergraduate book on Harmonic Analysis:







Question 1: What is the relationship between this description of a "dual basis" and this more traditional one?
Question 2: To check my understanding, let $N = 2$ and consider the basis $\{(2, 0), (0, 2i) \}$ of $\mathbb{C}^2$. Is its dual basis $\{1/2,0), (0, i/2)\}$ in this setting?

It is the same, under the canonical identification of $\mathbb C^N $ with its dual. The duality consists of writing every functional as $v\longmapsto \langle v,w\rangle $. 
So from the formula you have $$\langle w_k,v_j\rangle=\delta_{k,j}, $$ which makes $w_1,\ldots, w_N $ the dual basis.
To your second question: yes.
