# Why do we say a linear space is "over" a scalar field?

This terminology has puzzled me for a while and I haven't seen it actually discussed anywhere. Why does the language indicate relative positions of some space or operator and the objects they deal with?

• I wonder if it is related to the terminology of a ring $F[x]$ of polynomials ‘over’ the field $F$, and who was first responsible for these locutions.
– MJD
Jan 27, 2014 at 14:25
• Related would be: do they use something like "over" for this in other languages? Jan 27, 2014 at 15:48
• @GEdgar That's a good point - do we know if "over" is purely an Anglophone construct or does the notion exist in other natural languages as well?
– user98131
Jan 27, 2014 at 15:56
• @GEdgar (and Alexander Yasgar), it's also used in German, "Sei $V$ ein Vektorraum über $K$". Jan 29, 2014 at 21:27
• I really don't see why this should raise eyebrows at all. What could it be called instead that you couldn't ask this question about? A space... "around" a field? With? About? It had to be called something. Jan 30, 2014 at 11:42