Terminology for Substructures in Vector Spaces

[Mr Davis 97]

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."

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.

 

[fresh_42]

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."

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.

A vector space is a set in the first place. The additional structures turn this set into a vector space. A subset doesn't carry this additional structure (in general). A nail is a piece of iron with an additional shape. Some iron atoms can be part of the nail, without being one itself. Nevertheless would we call them a subset of the nail, and not a subset of some other set of iron atoms.

Also, isn't a vector space technically a quadruple (V, F, +, *) that satisfies certain axioms?

Yes.

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?

Yes, and this kind of shortcuts occur in many places. "Be $V$ a vector space" already tells us which structure it carries, so it's usually not needed to describe it in more detail. Only dimension and field would be helpful, and these are usually mentioned.

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.

Yes, as iron atoms don't have to be a nail, but we know that nails are iron atoms in a certain shape. The set is the collection of points we talk about. The distinction between such a set $V$ and a vector space $(V,\mathbb{F},+,\cdot)$ is done by calling it a vector space.

Of course you could be very rigorous and call the set $V$ the image of the vector space $(V,\mathbb{F},+,\cdot)$ under the forget-functor (which forgets the structure and is a functor between the category of vector spaces (e.g.) and sets), but this would blow up all texts without actually providing additional information. Therefore it's mostly left out.

 

[Stephen Tashi]

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.

That's a good point! One vector space consists of several sets. These include the set of vectors and the set of scalars. There are the various binary operations of vector addition, scalar addition, multiplication of scalars times scalers, mulitiplication of vectors times scalars - and a binary operation is a type of ordered set.

However, cultural tradition in use of mathematical language is that most mathematical structures are talked about as if they were a single set of things ( the "primary set", so to speak - although that's not standard terminology) and the other associated sets aren't mentioned as sets, but rather in terms of properties of that primary set.

For example, often books on group theory are careful use different notations for a group "$G$" as an algebraic structure and the set "$\{G\}$" whose members are the elements of the group. Books on vector spaces often aren't careful to distinguish between a vector space $V$ and the set $\{V\}$ of vectors. Your textbook is following a long tradition of ambiguous notation.