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This weekend we were discussing the strange locution
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The word “over” here means “the field of scalars of this vector space is…”. It would be less obscure to say “this is a vector space with real scalars” or “a general vector space with scalars from field F”. I don't know why we use the word “over”. Most “over” in mathematics suggests some sort of spatial intuition, or else refers to a particular notation in which something is over something else on the printed page. This is certainly not the second of these and I don't think it is the first either. Vector spaces have a very strong geometric flavor, but the scalars are not independent and they are certainly not under the rest of the space. I thought about it more and I think I remembered the first time I heard "X over Y" to describe a vector space. I was puzzled, but I don't think I was puzzled because of the spatial language. It was because the example was “a basis for the reals over the rationals”. I think there is no way anyone could understand what this means unless they had already heard “over” used this way or unless they were already familiar with the bizarre mathematical object being described. The real numbers can be considered as a vector space where the scalars are rationals, easy enough, but then if you want the vector space to have a basis you have to go to mathematical lala land to find it. Aha, I thought of another use of “over”: we can sum over the terms of a series or over the values of an index set; similarly in computer programming we loop over the elements of an array or we map over a sequence.
