Vector spaces versus subspaces

Question

I have learned that vector spaces satisfy the 8 properties including $(u+v)+w=u+(v+w)$ or $v+w=w+v$ , etc. but I'm now confused with the following question:

Are vector spaces closed under addition? In other words, if $a, b$ are in vector space $V$, is $a+b$ also in $V$?

Basically why I'm confused is because I've thought that this was a property for a 'subspace' of a vector space...

Answer 1

Closure under addition is a foundamental property for vector spaces and thus also for vector subspaces.

Answer 2

Usually, when you define a vector space you define addition to be a function of the type $+:V\times V\to V$ (some call this a binary operation on $V$). Since the codomain of $+$ is $V$, it is obvious that for all $v,w\in V$, $v+w$ is also in $V$.

Now, if you consider a subset $W$ of $V$, it is not necessarily true that for all $w_1,w_2\in W$, $w_1+w_2$ lies in $W$ (for it is only guaranteed to lie in $V$).

Answer 3

A vector space is closed under addition by definition.

A subspace of this is a subset that is also a vector space in its own right. So it too must be closed under addition.