There is a difference between the multiplication op in the Field of Reals and the Scaling op in the Set of Vectors. In the Field, the op is a closed mapping between two members within the same Set, whereas in the Set of Vectors, the op is a mapping between a member of the Set of Vectors and a member from the Set of Scalars (a different set entirely) that is closed within the Set of Vectors. The question arises: if we now let the Set of Vectors become one and the same with the Set of Scalars (i.e., the Field of Reals), do we indeed still have a Vector Space that is also a Field or do we just have a Field? If we just have a Field, is it still a full fledged Vector Space? I am uncertain of the answer as all Vector expressions would resolve to a single numerical value, collapsing the internal mechanical workings that make a Vector Space useful as a model for Physical processes.(adsbygoogle = window.adsbygoogle || []).push({});

Additional question: If the Field of Reals is duplicated so that we have a pair of twin Fields and we can, in theory, keep them entirely separate, can we answer the above question by saying "we have a Field of Reals as a Vector Space over a twin Field of Reals and we genuinely have a Vector Space in its conventional form"?

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# Is the Field of Reals in and of itself a Vector Space?

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