The definition of span In Linear Algebra by Friedberg, Insel and Spence, the definition of span (pg-$30$) is given as:

Let $S$ be a nonempty subset of a vector space $V$. The span of $S$,
  denoted by span$(S)$, is the set containing of all linear
  combinations of vectors in $S$. For convenience, we define
  span$(\emptyset)=\{0\}$.

In Linear Algebra by Hoffman and Kunze, the definition of span (pg-$36$) is given as:

Let $S$ be a set of vectors in a vector space $V$. The subspace
  spanned by $S$ is defined to be intersection $W$ of all subspaces of
  $V$ which contain $S$. When $S$ is finite set of vectors, $S =
\{\alpha_1, \alpha_2, ..., \alpha_n  \}$, we shall simply call $W$ the
  subspace spanned by the vectors $\alpha_1, \alpha_2, ..., \alpha_n$.

I am not able to understand the second definition completely. How do I relate "set of all linear combinations" and "intersection $W$ of all subspaces"? Please help.
Thanks.

Let $S$ be a non-empty subset of a vector space $V$.  The the set of all linear combinations of finite sets of elements of $S$ is called the linear span of $S$ and is denoted by $L(S)$.
