I have seen the following conventions, e.g.

  1. We define $K=\mathbb C$ ...
  2. We define $K$ to be $\mathbb C$ ...
  3. We define $K:=\mathbb C$ ...

I prefer number 3, because it is concise and it is clear what is defined to be what, similarly to a programming language when one writes $K\leftarrow\mathbb C$.

But is this in fact a commonly used notation or only used by a few mathematicians?

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    $\begingroup$ Sort of common, but not the most common. $\endgroup$ – Git Gud Aug 4 '13 at 14:57
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    $\begingroup$ I use the := thing constantly, and I learned to do so more than 26 years ago in undergraduate school. I think it is pretty simple and handy. $\endgroup$ – DonAntonio Aug 4 '13 at 14:57
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    $\begingroup$ A definition is a movement of thought that has a direction. Therefore one should use a sign for it that doesn't suggest symmetry, as the naked $=$ does, or even worse the symbol ${{\rm def}\atop=}$. $\endgroup$ – Christian Blatter Aug 4 '13 at 15:07
  • $\begingroup$ @ChristianBlatter Beautiful typographycal argument. $\endgroup$ – Git Gud Aug 4 '13 at 15:08
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    $\begingroup$ @ChristianBlatter: I agree that $:=$ is better than $=^{def}$, but IMHO either is preferable than the naked $=$. $\endgroup$ – Jakub Konieczny Aug 4 '13 at 15:13

If the sentence has the word "define" in it, then $:=$ is redundant. Redundancy isn't always bad, per se.

A mathematicians could also write it as "If $K=\mathbb C$ then $\dots$." In that sense, it isn't a definition. In computer programming, you have assignment, but you can reassign, so you get code like $x:=x+1$. That should never happen in mathematical writing, except when writing about programming/algorithms.

How do you deal with the case "$K=\mathbb C$ or $K=\mathbb R$?" That's one that often comes up, and it isn't so much a definition as a condition. And I think that's the real distinction - something what looks like a definition is often really a condition.

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    $\begingroup$ I'd say redundancy is very good in general, if it doesn't come at too high a price. Adding an extra ":" is not much trouble, and the sentence suddenly becomes easier to understand at first glance. $\endgroup$ – Jakub Konieczny Aug 4 '13 at 15:12

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