this post was submitted on 23 Apr 2026
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Also isn’t an infinite dimensional sphere practically hollow?
(If you were to integrate the sphere to calculate volume like you do for lower dimensional ones, you would sum the volume of shells—which is just their surface area times a thickness—making it up. With infinite dimensions, each shell becomes infinitely larger than the preceding shell no matter how fine you make the slices. This means the largest shell contains basically all the volume.)
This reasoning is pretty weird, but the conclusion is basically right. That is, there is absolutely no way to extend the conventional notion of volume to Rinfinity, which is basically what most people would imagine is the infinite equivalent of our dimensional space. Edit: what I mean by Rinfinity is a bit ambiguous, but let's say for the purpose of a hypersphere we want something like l^2 hilbert space to ensure no vectors with infinite length appear, then we have a separable space and the proof is complete.