To be honest, this is false too. You need functions with values that can be multiplied by scalars. If you just have ordered set, then you can't account for this

To be honest, this is false too. You need functions with values that can be multiplied by scalars. If you just have ordered set, then you can't account for this constant from the definition. Alternatively, you could say that values of the function need to be from a normed vector space (but I'm not sure if it can be extended furthermore. Maybe normed C-module?)