Suppose that $V$ is a vector space over $\mathbb{F}$, with $\dim V=n$. Let $H$ and $K$ be subspaces of $V$. Then $[H,K]$ is defined to be the set of all linear combinations of the commutators $[h,k]$ with $h\in H$ and $k\in K$. Or more concisely, $[H,K]=\langle\{[h,k]:h\in H, k\in K\}\rangle$.
I am struggling to understand the above definition of $[H,K]$ and make sense of what the elements of $[H,K]$ look like. Rewriting, I think that $$[H,K]=\left\{\sum_{i}\lambda_i[h_i,k_i]:h_i\in H\,, k_i\in K\,, \lambda_i\in\mathbb{F}\right\}\,.$$ But it doesn't seem obvious to me why this sum is finite. The basis of $V$ is finite, but there are infinitely many elements of $V$ (and hence $H$ and $K$).
I guess my questions are: what do the elements of $[H,K]$ look like explicitly and why is the sum of the commutators finite?
This is a question of vector spaces only. The sum is finite by definition. We are taking finite linear combinations of the basis elements of $U$, for any subspace $U$ in a vector space $V$. Even if $V$ is an infinite-dimensional vector space, linear combinations for subspaces are always finite sums by definition. We do not have convergence questions here.