Confusion on Summation Notation in Lagrange's Identity I'm working on a proof (the proof of Lagrange's Identity), but it includes a sum notation I'm not familiar with:
$$\sum_{1\le k\lt j\le n} (a_kb_j-a_jb_k)^2$$
I would appreciate any explanations of what this is saying, specifically in regards to the inequalities below the sigma.  Thanks!

${1\leq j<k\leq n}$ is the domain of the operator.   We sum the terms for all integer values of the bound variables, $(j,k)$, where this domain holds true. 
This is sometimes more convenient than the double sum notation with which you might be more familiar.
$$\sum_{1\leq j<k\leq n} (a_k b_j −a_j b_k )^2
\\ ~=~ \sum_{j=1}^{n-1}\sum_{k=j+1}^{n}(a_k b_j −a_j b_k )^2 
\\ ~=~ \sum_{k=2}^{n}\sum_{j=1}^{k-1}(a_k b_j −a_j b_k )^2 $$
