I'm confused about expanding the multivariate summation notations. It seems that following three equations are identical. However, I do not understand how these summation become equivalent statements.
It will be helpful if any of you can provide comments or simple expansion with N = 2 in order to follow below statements.
1) $ \sum_{i\neq j}^{N} a_i a_j$
2) $ \sum_{} \sum_{i\lt j}^{N} a_i a_j$
3) $ \sum_{1\le i\lt j \le N}^{N} a_i a_j$
Thank you
In all three each of $i,j$ are somewhere in between $1$ and $N$ inclusive. The extra information mentioned at the bottom of the sum is meant as additional restrictions on $i,j$ for the desired sum. This makes it clear that (2) and (3) are the same, since if $i,j$ are in the range and not equal, one of them is less than the other. [So(2) explicitly says $i<j$ and (3) just inserts the range "from $1$ to $N$ onto that.]
Now for (1) it will be the same provided one interprets $i \neq j$ as meaning that $i,j$ are taken to range over the subsets of $\{1,2,...,N\}$ which have size 2. For any such subset, one of $i,j$ is less than the other, determining a unique term.
I have seen the notation of sum (1) used in cases where one is summing over ordered pairs $(i,j),$ which (for N>1) would double the value of that sum compared to the other two.