Is it incorrect to use the subset notation with vector subspaces? Why?

Question

Let $V$ be a vector subspace of $\mathbb{R}^{n}$, and let $U$ be a subspace of $V$. The vector spaces and subspaces are ultimately sets of vectors, then why is everyone so reluctant to use the notation $U \subset V$?

Answer 1

In many different categories (e.g. vector spaces over a fixed field, groups, modules over a fixed ring, etc.), there is an underlying set for each object and some algebraic structure is preserved under each morphism.

So, in vector spaces, it's true that we can write $U \subset V$, or $U \subseteq V$ if we're not insisting on proper containment, but without proper context, this notation doesn't necessarily denote that $U$ is a linear subspace of $V$, just a subset of vectors. The only notation I've seen for this is $U < V$ (or $U \leq V$) to connote that $U$ is a subobject of $V$ in its given category.