What does it mean when you have Span ($s$) when $s$ is a vector space?

Question

As far as I know usually Span is noted with the finite set of vectors with which I can span the whole vector space.

For example in $R2$ obviously this is a span $Sp((1,0) , (0,1))$.

But I have a hard time understanding what it means when you put the vector space itself in the closure.

I have seen it quite a lot in questions when instead of writing a finite set of vectors they put in closures the vector space or subspace. Maybe they mean that the basis of that vector space should be the span? I am not sure. Can you help. By just pointing out the meaning of it? Thank you.

Answer 1

It just means look at the subspace generated by some finite set of vectors.

For example, if $S=\{(1,1),(2,2)\}$, then the span of $S$ is a subspace $U \subset V$ consisting of formal linear combinations $U=a_1(1,1)+b_2(2,2)$ with $a_1,b_2 \in \mathbb R$. If $S=\{(1,0),(0,1)\}$, then it just happens that the span of $S$ is all of $V$, since any element of $V$ can be written in the form $a_1(1,0)+a_2(0,1)$.

Answer 2

The span of a subset of the vector space is defined in exactly the same way, regardless of how many elements are in that subset. Namely, if $V$ is the vector space and if $X$ is a subset of $V$ then $\text{Span}(X)$ is the set of all elements of $V$ which can be written in the form $$a_1 x_1 + ... + a_K x_K $$ where $a_1,...,a_K$ are scalars (e.g. real numbers) and where $x_1,...,x_K$ are vectors (i.e. elements of $V$).

So if you take the span of $V$ itself, you'll discover, by applying the definition, that $\text{Span}(V) = V$, for a very simple reason: for each $v \in V$ you can take $K=1$, $a_1=1$, and $x_1=v \in V$, and you'll get $$v = 1 \cdot v = a_1 \cdot x_1 $$

Answer 3

The span of a set with possibly infinitely many elements is defined to be the set of all finite linear combinations of that set. Hence as mentioned in the comments, for example $\operatorname{span}(V) = V$ for a vector space $V$.