The definition of a vector space doesn't explicitly include closeness under addition and multiplication.
Is there a proof that shows or disproves it?
[Math] Are all vector spaces closed under addition and scalar multiplication? If so, why
linear algebravector-spaces
Best Answer
Now that you have told us what your question is about, I want to direct your attention towards the associativity axiom, which says that for arbitrary vectors $u,v,w$, $u+(v+w)=(u+v)+w$. This implicitly assumes that the vector space is closed under addition of vectors, because you can't apply the addition on elements not belonging to the vector space. Similarly, the axiom $a(bv)=(ab)v$ assumes closure under scalar multiplication.
This page should ease your worries: http://mathworld.wolfram.com/VectorSpace.html . Read the very very first line.