I was introduced to vector spaces recently, and when considering that they must be closed under multiplication.
$$c\mathbf{U} \in \boldsymbol{H} \text{ given } \mathbf{U} \in \boldsymbol{H}$$
What is the domain of $c$? Is it any element of any vector in $\boldsymbol{H}$?
We have primarily been talking about vectors in $\mathbb{R}^n$ so it has been handwaved as "any real number". But if it was an element, does that imply that you could define a vector space over $\mathbb{Q}^n$, $\mathbb{Z}^n$, or $\mathbb{N}^n$?
How does that relate to polynomial spaces or the vector space of all real-valued functions?
Best Answer
A vector space is a triplet $(V,F, \psi)$ where $V$ is a commutative (Abelian) group and $F$ is a field and $\psi$ is a special kind of function from $F\times V$ to $V.$
Before going further, a few standard conventions:
$+$ is used for the group-operation of $V$ and also for the addition-operation of $F.$
$ 0$ is used for the group-identity of $V$ and also for the additive-identity of $F.$
$1$ is used for the multiplicative-identity of $F.$
$fv$ is used for $\psi(f,v).$
The special properties of $\psi$ are:
If $f_1,f_2\in F$ and $v\in V$ then $(f_1+f_2)v=(f_1v)+(f_2v)$ and $f_1(f_2v)=(f_1f_2)v.$
If $f\in F$ and $v_1,v_2\in V$ then $f(v_1+v_2)=(fv_1)+(fv_2).$
If $f\in F$ and $v\in V$ then $[\,fv=0$ iff ($f=0$ or $v=0)\,].$
If $v\in V$ then $1v=v.$
This is called a vector-space over $F.$ But it is very common to refer to $V$ as the vector-space.
In some contexts $vf$ is defined to be $fv$ when $f\in F$ and $v\in V.$