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Say, I have the following sentence: "The Gaussian Sobolev spaces are as expected defined as follows" and then you do a definition environment after. The question is: how do you end "as follows"? With a period, or a colon?
I'd like to know if there is a strict English punctuation rule for this sentence.
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If the definition follows immediately, I'd use "as follows:" with a colon.
This is defined as follows:
(definition follows immediately)
If the definition doesn't follow immediately, but after something else, I'd use "below." with a full stop.
This is defined below.
(Some further discussion)
...
(Definition is reached at some point)
This isn't grammar; it's a matter of style. But "X is defined as follows" indicates that the definition follows (immediately); "X is defined below" indicates that you will find it if you carry on reading.
"Below" could work for both. But using a colon indicates "follows now" and a full-stop indicates "follows later".
Punctuation isn't a matter of grammar, but, as R L Trask explains in his ‘Guide to Punctuation’,
The colon is used to indicate that what follows it is an explanation or elaboration of what precedes it. That is, having introduced some topic in more general terms, you can use a colon and go on to explain that same topic in more specific terms.
So, yes, use a colon after follows, as you have done after your first six words.
