Norm induced by the complex dot product

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Ciro Santilli 34 Updated 2024-10-13 Created 1970-01-01

Given:

x = \sum_{k = 1}^{n} a_{k} + b_{k} i \in C^{n}, a_{k}, b_{k} \in R

the norm ends up being:

∣ x ∣ = \sum_{k = 1}^{n} a_{k}^{2} + b_{k}^{2}

E.g. in

C^{2}

∣ (2 + 3 i, - 1 + 5 i) ∣ = 2^{2} + 3^{2} + (- 1)^{2} + 5^{2} = 4 + 9 + 1 + 25 = 39