One of the more commonly used terms in linear algebra, and other areas that make heavy use of linear algebra, is that of a subspace. This term, I’ve been teaching a course on linear algebra, and we’ve gotten to the chapter on subspaces. Frequently, I end up saying X is a subspace of Y or some such thing. It seems to me that much of mathematical notation comes about because certain phrases and words are repeated so often that it greatly simplifies [1] writing if there is notation for them.

I’ve said this so frequently in class, and had to write it, that I’m beginning to wish that there were notation for subspace, yet, to the best of my knowledge, there isn’t. “X is a subspace of Y” is analogous to both “x is less than y” or “A is contained in B”/”A is a subset of B”. These three relations all fit in the framework of partially ordered sets. There is notation for the latter two. Why haven’t we come up with a corresponding notation for subspace? Or have we, and I just didn’t get the memo? (If I ever teach an algebra course, I’ll be complaining about the lack of notation for subgroup as well.)

[1] Well, it makes for less time writing, anyway. On the other hand, students don’t much care for new notation, so maybe it’s better to do without it.