6

I have seen people give the definition that a subspace is a vector space contained within a vector space. But is this definition actually accurate? Isn't this a special case, in particular the definition of a linear/vector subspace?

For example, I have seen hyperplanes described as subspaces, but they do not contain the zero vector (affine subspaces).

Shouldn't the correct definition of a subspace be something like this: A subset of a set from a space such that the structure of that space still holds on the subset.

In other words, my question is whether there is a general definition of a subspace.

Peter Mortensen
  • 670
Dylan Dijk
  • 323
  • 4
    A subspace of $V$ is a subset $W$ which is a vector space with respect to the same operations (addition and scalar multiplication) as $V$, when they are restricted to $W$. – Mark Nov 11 '22 at 17:49
  • Non-empty subset closed under the operations(suitably restricted) of the original space. – Lázaro Albuquerque Nov 11 '22 at 17:50
  • Note: I think that the question is great but that the title is kinda weird. – BCLC Nov 11 '22 at 18:55
  • Note that any vector space can be construed as an affine space (essentially by "forgetting the special role of the zero vector") and that if one does this, each hyperplane is exactly a sub-affine-space of the original space. So the construction that gets from a vector space to a sub-affine-space is quite analogous to the construction that gets you from a vector space to an arbitrary subset: forget about some of the original vector space structure and then consider a subobject of the resulting, less-structure object. – MJD Nov 11 '22 at 19:09
  • 1
    "A subset of a set from a space such that the structure of that space still holds on the subset." That is almost exactly one of the definitions for a subspace that I was taught. Or maybe it’s not a full definition, but a way to understand subspaces. – Todd Wilcox Nov 12 '22 at 07:25

2 Answers2

8

There's not enough words in the language for all the things we want to say in mathematics.

So, words get re-used.

It's very common in mathematics for the the same word to be used for two concepts in two different contexts, particularly when there is some intuitive connection between those two concepts, even if that intuition cannot be formalized.

So for example there are subspaces of vector spaces, and there subspaces of topological spaces. Those two subspace concepts are analogous; the comment of @LázaroAlbuquerque explains the analogy in some semi-formal sense. But, if there's no particular reason to formalize the analogy --- and in this case I don't think there is any reason --- then mathematicians will not bother trying to do so.

On the other hand sometimes there are reasons to try to formalize these analogies, and for that we have category theory. For instance, one might wish to argue that the concept of a monomorphism captures the general intuition of a sub-object of any mathematical object (more accurately, an embedding of one object as a sub-object of another object). If my understanding is correct this works reasonably well for subspaces in both the vector space category and the topological space category.

Lee Mosher
  • 120,280
5

No, you are exactly right.

Shouldn't the correct definition of a subspace be something like this: A subset of a set from a space such that the structure of that space still holds on the subset.

A term for

a vector space contained within a vector space (where the structures don't necessarily match

might be something like 'vector subset'.

There's a similar thing for groups in general:

For example, you can make a group $G$ of the set $[0,2\pi)$ and some operation (I forgot the specifics, but I believe it's described in this elementary introduction of adding angles) s.t. $G$ is isomorphic to the circle group. $G$ then is not a subgroup of $\mathbb R$. But since $G$'s set $[0,2\pi)$ is a subset of $\mathbb R$ and since both $G$ and $\mathbb R$ are groups, I like to think of $G$ as a 'group subset' of $\mathbb R$.

Differential geometry:

You can even distinguish submanifold from 'manifold subset'.

Or even just topology:

Topological subspace vs topological subset.

In category theory:

There's 'sub-object', so maybe category theory also has 'object subset' ? XD

BCLC
  • 13,459
  • 1
    One might add that in category theory one has the general notion of "subobject": a subobject of $A$ is any object $B$ for which there is a monomorphism $B\to A$. This covers all the cases one would want to, for example when the objects are topological spaces, vector spaces, groups, rings, fields, or even sets. – MJD Nov 11 '22 at 19:00
  • @MJD thanks. so 'subobject' and then...does category theory have 'object subset' ? XD – BCLC Nov 11 '22 at 19:01
  • 2
    You could define it in terms of the forgetful functor into the category of sets, but doing that would be philosophically the opposite of what category theory is all about. The whole point of category theory is to stop thinking of mathematical objects as being essentially sets with extra structure. – MJD Nov 11 '22 at 19:05
  • @MJD Ok I'll just take your word for it: 'object subset' is to do with something called 'forgetful functor' ? – BCLC Nov 11 '22 at 19:17
  • 1
    I am not sure what you mean by "object subset" so I can't say. – MJD Nov 11 '22 at 19:26
  • @MJD It's a generalisation of group subset: $(A,f)$ is a group and $(B,g)$ is a group and then $A \subseteq B$ or $A$ is in bijection with $C$ and $C \subseteq B$ but $(A,f)$ is not a subgroup of $(B,g)$ i.e. there doesn't exist a monomorphism or something then just a subset group. Make sense? – BCLC Nov 11 '22 at 20:16
  • So, for example, $\langle \Bbb Z, +\rangle$ is the integers under addition, and you consider the subset ${0, 1, 2, 3}$ with the operation of addition mod 4? – MJD Nov 11 '22 at 21:28
  • @MJD errrrr not a subgroup of course just a subset group...? – BCLC Nov 12 '22 at 07:11