I guess I was wondering if some of the Online Categorical People would turn up to say that this is some general principle about posets, analogous to the pigeonhole principle.

Matt is on the right lines concerning the “bonus question”, although my intended application is more pedestrian. I may get round to posting an update as a separate post.

]]>Bonus question: I think this is from some construction like the full group Cstar algebra. You are taking the supremum over a set of representations, and you want to argue that as the set is closed under infinite direct sums, supremums are attained. ]]>

If I want to talk about some dimension function on separable subspaces of some huge space (or whatever that is), it is better I simply talk about that, no need to introduce a new spooky h on some abstract poset. In a case where I can’t resist, I might add a remark: “note that all used was….”. Even when you *do* have another application coming up it is sometimes clearer to prove the first instance in the wild, and in the second application just note that the same proof applies. Finally, I am not sure that “cleaning the clutter” always makes for a cleaner exposition. To the contrary, sometimes the general idea is already clear in a certain instance, and everyone can see the generalization without needing to write it down.

Sometimes it turns out to be worthwhile to formulate more generally, but it’s hard to know that in real time.

]]>Thanks Matt – that seems to work.

Originally I was actually trying to reconstruct something like this argument, but somehow got myself mixed up when trying to use Sakai’s result. (Inevitable consequence of trying to dash off a blog post at 3am.)

]]>So in your case, if T* and V* were isometrically isomorphic, we could identify T with the normal functionals on V*, namely V (or the other way around) thus showing that T and V were (isometrically) isomorphic.

Notice that this argument is very much an “isometric” statement.

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