ACIT4330-Page/content/Definitions/Inner Product.md

Definition

(\cdot | \cdot) : V \times V \to \mathbb{C}

[!info]

V \times V \ni (u, v) \mapsto (u | v) \in \mathbb{C}

Such that (au + bv | w) = a(u | w) + b(v|w) and \overline{(u | v)} = (v | u)

[!example]-

(w | au + bv) = \overline{(au + bv | w)} = \overline{a(u | w) + b (v | w)} = \bar{a} \overline{(u | w)} + \bar{b} \overline{( v | w)} = \dots

and (v | v) \geq 0 and (u | u) = 0 \implies u = 0