# Using “adjunction” to refer to the act of taking adjoints of operators

I have an especially flabby terminology question.

How acceptable is it, in your opinion, to use the word "adjunction" to refer to the process of taking adjoints of operators on a Hilbert space?

An example of the kind of usage I'm thinking of might look something like

Therefore, the map $T \mapsto TS$ is continuous for each $S$, and continuity of left-multiplication follows by adjunction.

I always find myself wanting to write things like this, but only rarely see it used by other people. Is this terminology deprecated for some reason?

• I would also use it! – Berci Dec 16 '12 at 1:00
• It's absolutely etymologically correct, and it does fill something of a lexical gap. Yet I too have only ever seen the phrase "taking adjoints" in the context you have in mind. – Branimir Ćaćić Jun 10 '13 at 8:46

## 2 Answers

I’d advise against using it in this context. It’s etymologically correct, true; but it’s neither common nor totally transparent, so non-native speakers of English may be confused by it, and even native speakers may do a double-take. Since there’s a clearer and more usual alternative, “taking adjoints”, you should avoid tripping the reader up with language unnecessarily — help keep their concentration focused on the mathematics itself!

(However, in a different context — Category Theory — adjunction is a defined term in its own right, and by adjunction is very common in describing proofs.)

• Duly noted. Thanks for your perspective. – Mike F Jun 14 '13 at 18:51

Sure why not, right? Who's it going to hurt?