What is the meaning of ⊊? I have encountered this when referencing subsets and vector subspaces. For example, T ⊊ span(S) should mean that T is smaller than span(S)--at least from what I've gathered.
Is ⊊ a sort of ≤ or < but for subsets? What would that little crossing line mean? I have tried looking it up on the net but could not find an answer.
Thank you.
 A: If you look into some random books dealing with set theory (maybe as introductory to other subjects), you'll find that in some of them
$$
A\subset B
$$
means “for all $x$, if $x\in A$ then $x\in B$” (that is, “$A$ is a subset of $B$”).
In other books, you'll find the same relation denoted by $A\subseteq B$, whereas $A\subset B$ would mean that $A$ is a proper subset of $B$.
However, due to the notational conflict, somebody prefers to be as clear as possible and will write
$$
A\subsetneq B
$$
to mean that $A$ is a proper subset of $B$ (that is $A$ is a subset of $B$, but $A\ne B$). People who write $A\subset B$ for “subset, equality possibly happening” will probably use this notation for proper subsets.
Different traditions. When you see $A\subset B$, look in the initial pages to see what it is bound to mean. If there's no hint, try and infer from the usage.
A: Short answer:
$A\subsetneq B$ means that $A$ is a subset of $B$ and $A$ is not equal to $B$.

Long answer:
There is some confusion on mathematical textbooks when it comes to the symbols indicating one set is a subset of another.
It's relatively clear what the symbol "$\subseteq$" means. This symbol is more or less universally understood as the following:
$$A\subseteq B \iff \forall a: (a\in A: A\in B)$$
that is, $A$ is a subset of $B$.
The problem comes when we want to talk about the relation of "proper subset". That is, the relation described by the property
$$A\subseteq B \land A\neq B$$
Some textbooks use the sybol $\subset$ for this relation. That is, they define "$\subset$" as
$$A\subset B \iff A\subseteq B \land A\neq B.$$
The problem arises when some authors and mathematitians use the symbol "$\subset$" when they are talking about "any subset" instead of "proper subset". In other words, some authors use "$\subset$" in places where other authors would use $\subseteq$. This has caused the symbol "$\subset$" to become ambiguous, and authors tend to want to avoid it. To describe the relation "proper subset", they instead use the symbol "$\subsetneq$", which is less ambiguous.
TLDR: It's a mess, and because we don't want to use $\subset$, we use $\subsetneq$ to avoid ambiguity.
A: It means the space $T$ is a propper subspace of $span(S)$
