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?