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    $\begingroup$ What if "infinite-dimensional vector spaces with extra structure" don't occur more often per se, but there's so little general theory for infinite-dimensional vector spaces without extra structure that only the ones with structure are observed & studied. $\endgroup$ Commented Aug 16, 2023 at 14:16
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    $\begingroup$ I think one crucial point you're missing is that finite dimensional vector spaces implicitly come with extra structure. There is only one Hausdorff topology compatible with each finite dimensional vector space. All norms are equivalent etc. The topology is somehow baked in. The only difference is in infinite dimensions you must make a choice. $\endgroup$ Commented Aug 16, 2023 at 15:02
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    $\begingroup$ You might just as well ask "Why do infinite sets usually have extra structure?" $\endgroup$ Commented Aug 16, 2023 at 15:49
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    $\begingroup$ Re, you are definitely using an inner product when you speak of rotations! Otherwise, where do the angles come from? $\endgroup$ Commented Aug 16, 2023 at 16:51
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    $\begingroup$ Here's another perspective. From textbooks, we can get the impression that mathematical objects begin life as structureless sets, and acquire more structure as we add it. But in "real life," we usually start with examples, which come with oodles of structure, and much of our task as mathematicians is to strip away much of that structure in order to clarify certain logical relationships. From this perspective, the question isn't why infinite-dimensional vector spaces have additional structure. The question is why we can't strip away all the structure and still deduce the theorems we want. $\endgroup$ Commented Aug 17, 2023 at 14:50