# Admitting a unique fixed point in math

I've noticed that some of my professors seem to like to use the term admit in the following sense (from the contraction mapping theorem): "If F is a contraction from a complete metric space to itself, then it admits a unique fixed point."

I'm curious as to the word choice/usage of the term admit. It seems to be equivalent in usage, at least here, to 'has', but I see it in quite a bit of formal math from certain professors.

Does it have a particular connotation/meaning that has doesn't?

In mathematics I believe it means something more like

to allow or concede as valid

TFD.

• +1 So it does inside of mathematics also. As a mathematician, I can attest that this is current usage, dictionaries be damned. Feb 10, 2015 at 5:58
• I don't really understand this answer. The OP's sentence with admits sounds natural to me, whereas "the contraction F allows a unique fixed point as valid" sounds like gibberish. (And likewise with "concedes".) Feb 10, 2015 at 6:42
• @ruakh then don't say it that way. "If F is a contraction, then it allows/has/admits a fixed point.". Feb 10, 2015 at 12:50
• @Robusto: As I said, I already tried it. "The contraction F concedes a unique fixed point as valid" sounds like gibberish to me. Feb 10, 2015 at 16:16
• @Mitch: Re: "dictionary words aren't (always) one-to-one replaceable with their definitions": That's true. And in fact, even in cases where admit actually does mean "to allow or concede as valid", the latter phrasing could be a bit awkward. ("I conceded the force of his argument as valid" is rather strange.) But surely a dictionary definition, in order to apply to a given sentence, must have something to do with the usage in that sentence. (Other than, "it's a definition for a different sense of the same word". ;-) ) Feb 10, 2015 at 18:32

Yes. In mathematics, 'admit' is often used in statements where a mere 'have' would do the job quite well, and the connotation that this word can bring in is that it is nice or something of an achievement that the object has the property (i.e., the object admits the property).

For example, in proof theory, one often hears that a calculus 'admits' cut elimination (example usage). One might say 'has cut elimination' and it would convey the same fact (and is also often used), but 'admits' conveys the attitude of the speaker/community to cut elimination in that it is a very useful property for a calculus to have, it is inobvious that the calculus has it, and it is sometimes difficult to achieve. In a sequent calculus, cut is one of the derivation rules. Now proofs using cut can be nasty. Cut elimination is a procedure whereby a proof in that calculus that can in principle employ all of the derivation rules is transformed into another proof that makes do without the cut rule. Cut-free proofs tend to be nicer because they have the subformula property: each formula occuring in the proof is a subformula of the formula (sequent) to be proved. This facilitates proof search.

I believe that using 'admit' in your example is motivated analogously. It actually conveys two basic facts: that a fixed point exists and that it is unique. The Wikipedia entry on contraction mapping has the following:

A contraction mapping has at most one fixed point. Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems...

The existence of a fixed point, in this case, would be the strong or desirable property for the mapping to have.

I think the connotation is that something admitted is something that is constructed, rather than emerging naturally as an inherent property.

• That's an opinion, right?
– Kris
Feb 10, 2015 at 5:55
• Yes, it's an opinion. Feb 10, 2015 at 10:44

"Admission" in this sense connotes an idea that is conceded by way of a proof.