Smallest Subspace of V In the book 'Linear Algebra' by Hoffman, the following is noted

From Theorem 2 it follows that if $S$ is any collection of vectors in $V$, then there is a smallest subspace of $V$ which contains $S$, that is, a subspace which contains $S$ and which is contained in every other subspace containing $S$.

Isn't that smallest subspace just $S$?
 A: The comment of Thomas Bladt is absolutely correct, I'm writing an example so that maybe you can understand better the situation.
As a vector space I'm considering $V=\mathbb{R}^4$ (with the usual sum and scalar structure). Let $S=\lbrace v_1, v_2, v_3\rbrace$ where 
$$v_1=\begin{pmatrix}1\\0\\0\\0
\end{pmatrix}, v_2=\begin{pmatrix}0\\1\\0\\0
\end{pmatrix}, v_3=\begin{pmatrix}1\\1\\0\\0
\end{pmatrix}.$$
Now you can see that $S$ is just a collection of vectors, in this case finite. $S$ is absolutely not a subspace of $V$ as for example $v_2+v_3$ or $44\cdot v_1$ are not in $S$.
The sentence that you wrote above claims that there is a smallest subspace $W$ of $V$ that contains $S$. What is $W$ in this example? As an exercise you can check that $W=\text{span}(v_1, v_2, v_3)=\text{span}(v_1, v_2)=\mathbb{R}^2\times\lbrace 0\rbrace^2$ and also that $W$ is the smallest subspace containing $S$. Concretely this means that you have to check that if $W'$ is any subspace containing $S$, then $W\leq W'$.
