I'm trying to understand the definition of tensor product between two vector spaces, but in some demonstrations of the existence of the tensor product there is the following assumption: "Let $U$ and $V$ be vector spaces over a field $K$ and $\langle U\times V\rangle $ be a vector space over the same field whose base is $U\times V$". What I don't understand is why there's a vector space $\langle U\times V\rangle $ whose base is the set $U\times V$. By the way some write this assumption it seems to me that given a non-empty set $B$ is possible to construct a vector space $X$ whose base is the set $B$.
Something I found that may be related to my question is about free vector space which is described in the following link: https://math.berkeley.edu/~shiyu/s15capstone/materials/Capstone_Course%20(7).pdf
But, given a vector space $V$ over a field $K$, I don't know how to construct a free vector space whose base is the set $V$.
Please explain as detailed as possible how to construct a vector space whose base is a given set. I would like an explanation to finally understand the construction of a tensor product using this idea of constructing an appropriate vector space.
Given some set $B$, consider the set $F_B$ of all functions $f\colon B\to K$ such that $f(b)\not=0$ for at most finitly many $b\in B$. This is a vector space over $K$, in fact a subspace of the space $K^B$ of all mappings defined on $B$ with values in $K$. A basis of $F_B$ is given by the mappings $c_b, b\in B$, where $c_b(b'):=0$ for $b'\not=b$ and $c_b(b):=1$. Finally one may identify $b$ with $c_b$.