If a subset $W$ of a vector space $V$ is a subspace of $V$, I want to show that $\operatorname{span}(W)=W$.
But is it possible to define $\operatorname{span}(W)$? $W$ can either be a finite or infinite set and according to what I know $\operatorname{span}(W)$ can only be defined on a finite set $W$.
Is there any problem with my argument?
On
Span is defined for all sets of vectors: if $X\subseteq V$, then $Span(X)$ is the set of vectors which can be written as linear combinations of elements of $W$.
Now, in each specific linear combination, only finitely many vectors from $X$ can be used, so in that sense span is "about" finite sets; but $Span(X)$ is still defined, and will in general be wildly different from $Span(F)$ for any finite $F\subseteq X$. (Example: let $X$ be any infinite set of mutually transcendental reals, as a subspace of the $\mathbb{Q}$-vector space $\mathbb{R}$.)
The span of $W$ is all vectors produced from (finite) linear combinations of vectors in $W$. That is to say, $$ \operatorname{span}(W)=\left\{ \alpha_{1}w_{1}+\cdots+\alpha_{n}w_{n}\colon n\geq 1,\text{ }\{\alpha_{i}\}\subset K\text{, }\{w_{i}\}\subset W\right\} . $$ So yes, the span of an infinite set is well-defined. $K$ is the underlying scalar field (e.g. $K=\mathbb{R}$ if your scalars are real numbers).