I met in several sources (textbooks) phrase «Space can be constructed over any field». But it is always illustrated with linear space over scalar field (or sometimes over ring). Does it make any sense to talk about spaces over vector fields? What kinds of spaces are they? What about tensor fields? Can they spawn spaces as well? By the way Google search by «space over vector field» (quotes included) returns single hit, scientific paper which is too specialised for me to understand. And the second question related to the first one (I am not a mathematician, so I am sorry if my questions appear to be naïve). What is a vector field from the viewpoint of abstact algebra? In abstract algebra chapters, they mention several classes of set-based structures, namely, groups, rings and fields (scalar ones!), linear spaces, different algebras, but not vector fields. I am under impression that in terms of operations a vector field does have all characteristics of an algebra: addition, multiplication by scalar, vector product (although, dot product seems to be some additional constraint, does it?), but I am puzzled that I never saw it stated clearly in any text I checked.