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For example, when you have an expression like this,

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

what is this sum if one wrote it out?

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    depends on whether it is an infinite sum or finite sum. If it's finite it's just a sum of all $g^{kl}$ where $k,l$ ranges over finite values. If it's infinite sum, then it is an integral with respect to counting measure – 2017-01-31
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    Write out the sum over k so that each term still has an l index, then sum over l. You can do it the other way round too. – 2017-01-31

2 Answers 2

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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.

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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.