Well, my definition of span is the following:
Span:
Suppose a vector space $(V,+,\cdot)$, and
$$S = \{u_1,\cdots,u_n\}$$
(and $S$ is a subset of $V$, not a subspace)
$$[S]=:\cap_{w\subset V, w\supseteq S} W$$
In other words, $[S]$ is, by definition, the intersection of all $W$,
such that $W$ is a subset of $V$ and $W$ contains $S$.
So, if I want to find $[\emptyset]$, I need to look for all the subspaces that contains $\emptyset$. Well, every subspace contains $\emptyset$, because every subspace is a set. Therefore, I could imagine all diferente types of subspaces. Their intersection surely is gonna have the $0$ vector, because every subspace must contain it. But what guarantees that the intersection of the definition is gonna have only the zero vector? Agaun: I can imagine a lot of subspaces that contain $S$ (in this case, $\emptyset$), but also a lot of other elements. The definition says nothing about the space $V$.
I've found this proof, but I can't understand where did he took "smallest subspace possible", from the definition.
Also, in which way can we find the "basis for the empty set"?
Is this possible? My teacher said to us that we should find it. Is the basis for the empty set, the $0$ vector of $V$?
Best Answer
Note that the zero subspace, which is simply the set $\{0\}$, is one of the subspaces in your intersection and hence that intersection cannot have any vectors in it other than $0$.
Incidentally, as you have already noted that every subspace must contain $0$, the zero subspace, which only contains $0$, is the "smallest possible subspace", as Henno points out in your link.