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The axioms of the inner product are

1. [tex]\langle V|W\rangle = \langle W|V\rangle^*[/tex]

2. [tex]\langle V|V\rangle \geq 0\ \ \ \ \ 0 \ \ iff\ \ |V\rangle = |0\rangle[/tex]

3. [tex]\langle V|(a|W\rangle +b|Z\rangle ) \equiv \langle V|aW+bZ\rangle = a\langle V|W \rangle +b\langle V|Z \rangle[/tex]

Given that [tex]|V\rangle[/tex] and [tex]|W \rangle[/tex] can be expressed in terms of their basis vectors,

[tex]|V \rangle = \sum_i v_i |i \rangle[/tex]

[tex]|W \rangle = \sum_j w_j|j \rangle[/tex]

Shankar says "we follow the axioms obeyed by the inner product to obtain"

[tex]\langle V|W \rangle = \sum_i \sum_j v_i^*w_j\langle i|j \rangle[/tex]

I don't understand how this comes about?

thanks