What is $\sum_{k, l}$?

Question

For example, when you have an expression like this,

$\sum_{k, l} g^{kl}$,

what is this sum if one wrote it out?

Answer 1

It's usually shorthand notation for $$\sum\limits_{l}\sum\limits_k g^{kl} = \sum\limits_{k}\sum\limits_l g^{kl}$$

These sums are equal as long as there are finite indices.

Answer 2

Sometimes ranges are not specified in sums, it's given in context. Here, you need to sum over the indices $k$ and $l$ in their respective ranges. This would be a double sum.