Can Complex Euclidean Space Be Defined?

[gotjrgkr]

Homework Statement

Euclidean space is the set of n-tuples with some operations and norm.
I suddenly wonder if complex euclidean space can be defined.

Is it also defined?

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Certainly: $C^n$ is the set of n tuples of complex numbers with the norm defined by $\sqrt{\sum_{i=1}^n z_nz_n^*}$ where the "*" indicates the complex conjugate.

But unless you have some kind of additional structure the uses the algebraic properties of complex numbers, that is identical to $R^{2n}$.