Orthonormality Properties of the Dual Basis in $\mathbb{C}^N$

Question

1 Context

I'm working on an undergraduate textbook on Harmonic Analysis. The notational buildup for a problem I'm having is long but fairly straightforward and standard:

2 Problem

Attempt:

1. First note that

   $$ v_k = a_0 v_1 + \ldots + a_k v_k + \ldots + a_N v_N = 0 v_1 + \ldots + 1 v_k + \ldots + 0 v_N $$

   so that $a_j = 0$ if $k \ne j$ and $a_j = 1$ if $k = j$.

2. But

   $$ \langle v_k, w_j \rangle = a_j $$

   which from (1) shows that

$$ \langle v_k, w_j \rangle = \delta_{j,k} $$

I'm skeptical that this argument is right though since it doesn't depend on $w_j$. Moreover, how does one show that

$$ \sum_{j=1}^N \langle v, w_j \rangle v_j = \sum_{j=1}^N \langle v, v_j \rangle w_j? $$

Answer 1

The first part of this exercise is already given by your post, as the coordinate of each $v_k$ under the basis $\{v_1,\cdots,v_n\}\subset \mathbb C^N$ is exactly $[\delta_{kl}]_{l=1}^N$.

For the remnant part of this exercise, I would like to use the notion of dual space and dual basis in linear algebra, but you can avoid these notions and prove the exercise in the same manner with no difficulty. Let us denote $$ v^j:=\langle -,w_j\rangle,\ w^j:=\langle -,v_j\rangle. $$ It is clear that $w^j$ and $v^j$ so defined are linear functionals over $\mathbb C^N$, i.e., elements of the dual space $(\mathbb C^N)^*$. The orthonormality proved in this exercise indicates that $\{v^j\}$ and $\{w^j\}$ are exactly the dual bases of resp. $\{v_j\}$ and $\{w_j\}$ in $(\mathbb C^N)^*$. Now for each $v\in \mathbb C^N$, one has $$ v=\sum_{j=1}^Na_jv_j=\sum_{j=1}^Nb_jw_j $$ for some $a_j,b_j\in\mathbb C$ (since $\{v_j\}$ and $\{w_j\}$ are both bases of $\mathbb C^N$), and then with $v$ acted by resp. $v^j$ and $w^j$, it can be seen that $$ a_j=v^j(v)=\langle v,w_j\rangle,b_j=w^j(v)=\langle v,v_j\rangle. $$ Thus it follows that $$ v=\sum_{j=1}^N\langle v,w_j\rangle v_j=\sum_{j=1}^N\langle v,v_j\rangle w_j. $$