My textbook gives the following definition: "A subset W of a vector space V over a field F is called a subspace of V if W is a vector space over F with the operations of addition and scalar multiplication."(adsbygoogle = window.adsbygoogle || []).push({});

I understand the definition, and subspaces in general, but am a little confused when it comes to wording. For example: "A subset W of a vector space V..."

How can a set W be a subset of a vector space V? I thought that sets could only be subsets of other sets, not of other objects like spaces.

Also, isn't a vector space technically a quadruple (V, F, +, *) that satisfies certain axioms? The set V is just one part of the structure as a whole. Is it just shorthand to call the structure by its set V, even though that is technically not correct? It seems that it could possibly get confusing, as the object V could simultaneously be a set and a vector space, even though those are two different, but related, objects.

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# I Terminology for substructures

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