Is there a proper notation for denoting subspaces? For example, if $U$ is a subspace of some vector space $V$. I would usually just write "the subspace $U \subseteq V$" but I'm wondering is there is a proper, more compact, notation.
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$\begingroup$I've seen $U \leq V$ and $U < V$ being used to denote subspaces and proper subspaces respectively, but these aren't common enough to be used without explicity specifying their meaning first.
Another way that at least one text book I've red used was to reserve certain letters for certain types of things. For example, $U,V$ would always be subspaces, $K$ would always be a compact subspace, $A,B,C$ would be plain subsets, and so on. Again, you'd of course need to explain your notation before using it.
I personally believe that custom notational conventions often make mathematical texts harder to read, not easier. If requires to reader to memorize your notational conventions before being able to work with the text.
$\endgroup$ $\begingroup$There is no special notation for that. In general in algebra there is no special notation for substructures. The exception is group theory, where $H\le G$ is sometimes used for "$H$ is a subgroup of $G$", and $H\lhd G$ to (much more commonly) indicate that "$H$ is a normal subgroup of $G$".
$\endgroup$ 2 $\begingroup$I would add a categorical way to write this :
$U \hookrightarrow V \in \mathcal{V}ect $
This mean V is a vector space (in the category of vector spaces ) and $U$ have an injection (monomorphism) to $V$ .
Since we put $\in \mathcal{V}ect $ this is clear that the monomorphism need to be a monomorphism of vector spaces and then U need to be a vector space
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