I'm writting an article, and I've to recall some properties of operators I'm considering for the development of some theoretical aspects. I was wondering, is it correct to use the "notation"
"Property X: some fancy and awesome property of an operator"
or should I use directly "Theorem X" without given the actual proof?
Thanks in advance!
On
I can think of three situations.
$(1)$ The property is well-known. The definition is included in every textbook on the subject. Most textbooks devote time to proving several equivalent forms of the definition.
In that case your readers probably saw the definition before. But you want to include tit anyway as a reminder. Maybe you want to select which equivalent version is most useful to your article. You should write something like this: (Note the use of italics to distinguish the relevant technical term)
. . . . The next few results apply only to compact operators. Recall the definition.
Definition: The operator $A: V \to W$ is called a compact operator to mean the image of every bounded set is relatively compact.
Now assume $A$ is an arbitrary compact operator. It's easy to see that. . .
$(2)$ The property is not so well known. It was invented by someone else and published perhaps in one or two places.
In that case your readers are probably unfamiliar with the definition. You need to include it and mention the inventor, along with the publication where they first gave the definition:
. . . . For the purposes of this article, we consider only operators which are supercompact in the sense defined by Walter [2].
Definition: The operator $A: V \to W$ is called a supercompact operator to mean the image of every bounded set is super relatively compact.
Observe this is a stronger condition than demanding the image be relatively supercompact. . .
$(3)$ No one has ever heard of the property before. You invented it yourself and this is the first time publishing.
In that case you need to devote some time to what lead you to the definition and why it will be useful.
. . . . This reasoning naturally leads to the definition.
Definition: The operator $A: V \to W$ is called a superbcompact operator to mean the inverse image of every bounded set is superbly relatively compact.
To see what this definition means consider the following example. . . .
Either would be acceptable, though of course in both cases you should give a citation for the proof if possible. Personally, I'd choose Property over Theorem, because Theorem is usually used for "key points of the paper". It's generally good style to reserve Theorem for the main things you want the reader to take away, rather than the supporting material. I've also seen Fact used in this situation - it sort of implicitly suggests "this is true, it's not easy to prove, and I'm not going to give you the proof".