What does it mean for a subset of a vector space to be equal to its span? I came across a question in the textbook Linear Algebra by Stephen H. Friedberg:

Show that a subset of $W$ of a vector space $V$ is a subspace of $V$ iff $span(\{W\})=W$ (Exercise 1.4.12)

The forward proof itself is quite simple, since if $span({W})=W$, then $W$ is a subspace of $V$ since the span of subset of a vector space is always a subspace of the vector space; however, I am not able to understand what it means for the span of a subset to be equal to the subset itself.
Span by definition is all the possible linear combinations of the given vectors, that is, given the set of vectors: $\{u_1,u_2,...,u_n\}$ and constants $a_1,a_2,...,a_n \in F$, span is defined as follows:
$$span(\{u_1,u_2,...,u_n\})=a_1u_1+a_2u_2+...+a_nu_n$$
Now, let $W=\{u_1,u_2,...,u_n\}$. To say that $span(W)=W$ would mean the following:
$$\{u_1,u_2,...,u_n\}=a_1u_1+a_2u_2+...+a_nu_n$$
Which to me does not make any sense, how can a set equal an expression of the sum of linear combinations?
For context, I am a beginner to Linear Algebra and I am self studying the subject, so please excuse any inconsistencies in my mathematical notation/writing. It would also be really helpful if in such a case you could correct me, thank you.
 A: $(\Rightarrow )$ Since $\text{span}(W)$ is “smallest” subspace containing $W$ and $W$ is subspace of $V$, we have $\text{span}(W)=W$.
A: The span is not an expression, it is the set of all possible linear combinations so you would get
$$\mathrm{span}\lbrace u_1, \ldots, u_n \rbrace = \left\lbrace a_1 u_1 + \cdots + a_n u_n \mid a_1, \ldots, a_n \in F\right\rbrace$$
Moreover, the set $W$ is not necessarily finite. In general,
$$\mathrm{span}(W) =  \left\lbrace a_1 w_1 + \cdots + a_n w_n \mid n \in \mathbb N^{\times}, \ w_1, \ldots, w_n \in W, \ a_1, \ldots, a_n \in F\right\rbrace.$$
And for exemple, $\mathrm{span}(\mathbb F^n) = \mathbb F^n$, and if $W \subset V$, then $\mathrm{span}(\mathrm{span}(W)) = \mathrm{span}(W)$.
A: 
however, I am not able to understand what it means for the span of a subset to be equal to the subset itself.

Consider for instance two linearly independent vectors $u,v$ in some $n > 1$ dimensional space $V$. Then, span$(u,v)$ is a $2$-dimensional subspace $W \subset V$. But note also that $W$ is a set itself, and its span can be considered. In particular, span$(W)$ is the set of all linear combinations of elements of $W$. Since $W$ is a subspace, any such linear combination will fall again in $W$. Hence,
$$\mathrm{span}(u,v) = W = \mathrm{span}(W)$$
(or really span$(W) \subset W$ but we trivially have $W \subset \mathrm{span}(W)$ and the two inclusions imply equality). So, it makes perfect sense to consider the span of a set and hopefully this example helps you see why if the set in question is a subspace then it equals its span.
