Is there a specific symbol for denoting a linear subspace like $\subseteq$ for denoting a subset? I was under the impression that this was a proper notation, but I was just corrected on this but did not get a proper explanation of what it should be. I get that it is definitely not the same, but what is the right symbol then?
So I am specifically asking for a linear subspace. Or does this not make any difference?
Some authors denote that as $\leq$. The inclusion $\subseteq$ of set $X$ form a partial order of the power set $\mathcal{P}(X)$ of a set $X$ and if $X$ is vector space then it's subspaces form the suborder as a subset of $\mathcal{P}(X)$. In order theory, there are many orders that use different symbols $\leq,\ \subseteq,\ \lesssim,\ \preceq,\ \unlhd$, etc.
Usually the $\subset$ symbol is used for sets. However, any symbol can be used as long as it is predefined or can be distinguished from the context. There are a myriad of orders, so there is no need to create other symbols for them.
Indeed a linear subspace is a subset of a given linear space for which the linearity properties are preserved but there is not a specific symbol for that.
In a compact form we could use $$U=\operatorname{span}\{v_1,\ldots,v_k\}\subseteq V$$ beeing $\{v_1,\ldots,v_k\} \subseteq V$ a basis for $U$ or more in general any subset of the linear space $V$ depending on the context we are dealing with.