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Not much here, but: predual.
There might be some confusion for the reader since the definition section links to ’duals’ in the sense of rigid monoidal categories (or of other categories with duals), but as you know that’s not how the word ’dual’ is used in Banach space theory. Indeed, the monoidal category of Banach spaces is not a category with duals in any of the senses listed (q.v.).
But it should be, we just don’t have the right sense listed there.
This seems a bizarre use of terminology, but I presume it’s standard. Wouldn’t something like “codual” make more sense? Or “left dual” vs “right dual”?
’Left’ and ’right’ don’t help if one wants a right, say, predual.
I presume it’s standard
It is for people who use the term! I’ve not seen it used outside the context of Banach spaces (cf. present discussions on von Neumann algebras), but it’s common usage there.
The immediate purpose for writing the page seems to be these discussions, and for that particular purpose I do not support renaming it. If instead “predual” refers to “duals” in any of its standard monoidal category senses (as at category with duals), then no doubt it reduces to something with a more standard name (e.g., a left predual of is just a right dual of ).
I’m going to try to make some changes at predual to reflect this state of affairs.
It seems to me that the whole point of preduals is when duals don’t have the perfect duality of a category with duals, so referring to that was probably just a mistake on my part. It’s when we have infinite-dimensional vector spaces, irreflexive Banach spaces, etc that we care about preduals.
I think the fundamental problem is using the word “dual” for an operation that is not involutive and doesn’t even have an inverse. That just feels all sorts of wrong to me. (-:
The word ‘dual’ is almost meaningless; a dual is just something that comes with the first thing, making a pair (Latin ‘duo’ = ‘two’). We are lucky that we almost have a single meaning for it! (The same can be said of ‘derivative’.)
Edit: spelling.
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