[drdeca]

> increasingly implausible numbers of orthogonal spatial dimensions in order to render them ‘geometric’.

Implausible how? “geometric” doesn’t mean “embeds nicely in 3D space”.

What’s wrong with talking about the angle between two L^2 functions defined on an interval? Geometric reasoning still works? If you take a span of two functions, you have a plane. What’s the issue?

[HPsquared]

In this case can people just prepend "hyper-" as in hyperplane etc? Hyper-line, hyper-angle. (Speaking as someone who has heard 'hyperplane' a few times but not others)

[klodolph]

No, that would be incorrect. A plane is 2D. If you have two functions, and take their span, you get a 2D plane. It is a regular, flat, 2D plane.

When people say “hyperplane” they are generally talking about something with more than two dimensions.

[drdeca]

At least when the ambient vector space is more than 3-dimensional, yeah. Specifically, a hyperplane generally refers to something with codimension 1.

(So, when the ambient vector space is finite-dimensional, the dimension of a hyperplane is one less than the dimension of the ambient vector space.)