How to interpret $\sum_{1 \le i , j\le n} $?

Question

I have issues with understanding the exact meaning of $\sum_{1 \le i , j\le n} $. For example, how do I have to read the following combination of symbols?

$$\sum_{1 \le i , j\le n} a_i b_i \langle u_i, u_j \rangle$$

Answer 1

It is to be interpreted as:

$$\sum_{j=1}^n\left( \sum_{i=1}^n a_i b_i \langle u_i, u_j \rangle \right)$$

or equivalently:

$$\sum_{i=1}^n\left( \sum_{j=1}^n a_i b_i \langle u_i, u_j \rangle \right)$$

Answer 2

HINT: Think about first what $\left(\displaystyle\sum_{1\leq i,j\leq n}1\right)$ means.

It is nothing but a truncated way of writing the double sum $\left (\displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n 1\right )$, which equals $n\times n=n^2$.

Compare this to something like $\left(\displaystyle\sum_{1\leq i\neq j\leq n}1\right)$, which is simply $ \displaystyle\sum_{i=1}^n \displaystyle\sum_{\substack {j=1 \\ i\neq j}}^n 1 =n(n-1)=(n)_2$,

or $\displaystyle\sum_{1\leq i<j\leq n}1=\displaystyle\sum_{i=1}^n \displaystyle\sum_{\substack {j=1 \\ j>i}}^n 1=\frac{n(n-1)}{2}={n\choose 2}$