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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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You seem to have answered your own quesiton after the first six words. –  Barrie England Jan 1 '13 at 13:59
    
Though the natural choice would be a colon, there are many situations where it is not recommended. One such exception could be where what 'follows' is an elaborate text, sometimes extending over more than a paragraph. Unless the "definition environment" this case is no longer than a sentence or at most two, it would be a good style practice to end the introductory with a period. –  Kris Jan 1 '13 at 14:14
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closed as not a real question by MετάEd, tchrist, cornbread ninja 麵包忍者, Kris, Hellion Jan 2 '13 at 19:39

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2 Answers

up vote 3 down vote accepted

This question featured in chat.

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".

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It seems the answer in the chat cited was put in a slightly different way. It was not quite about the colon v. period. Even so here in a way. –  Kris Jan 1 '13 at 14:18
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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.

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