Let $X$ be a (potentially infinite dimensional) vector space, and suppose that $x\in X$. Take the finite indexing set $I$, which corresponds to the basis of $X$. That is, we can find $a_i$ in $\mathbb F$, for $i\in I$, such that $\sum_{i\in I}a_i x_i=x$. Additionally, let $b_j$ in $\mathbb F$ for $j\in J$, where $J$ is another finite indexing set, and consider the product
$$x\cdot\sum_{j\in J}b_j=\sum_{i\in I}a_i x_i\sum_{j\in J}b_j=\sum_{j\in J}\sum_{i\in I}a_ib_j x_i,$$
where the order of summation doesn't matter, since all sums involved are finite.
My question: if $I=\emptyset$, then what does this product look like? Is it, since $I$ is empty and there is no $x_i$ which can be indexed by anything there, that $\sum_{i\in I}a_i x_i=0$, which would mean $x\cdot\sum_{j\in J}b_j=0$? Or, is it that $\sum_{i\in I}a_i x_i=1$ so that $x\cdot\sum_{j\in J}b_j=\sum_{j\in J}b_j$?
Well, the convention is that an empty sum, i.e., $I=\emptyset$, equals the zero element in the structure (vector space, ring, etc.), here zero vector, and - what's not so important for you - an empty product equals the unit element in the ring.