Clarification on definition of a basis

Question

Quick question; lets say that $S$ is a basis of $V$. I understand that this means all vectors in $S$ are linearly independent, and that every vector in $V$ is an element of $\text{span} \ S$.

Is it also required that every element of $\text{span} \ S$ is an element of $V$?

I've been getting different answers from different sources so I'm a bit confused.

Thanks ahead of time.

Answer 1

$S$ is a familly of elements of $V$, so its span is in $V$.