Real coordinate space (${\mathbb{R}}^n$)

Important 4D spaces:

Simulate it. Just simulate it.

The complex dot product is defined as:

E.g. in ${\mathbb{C}}^1$:

We can see therefore that this is a form, and a positive definite because:

Just like the usual dot product, this will be a positive definite symmetric bilinear form by definition.

${\mathbb{R}}^n$ with extra structure added to make it into a metric space.

The identity matrix.

Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.