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I'm familiar with the advice by Geoffrey Hinton, "To deal with a 14-dimensional space, visualize a 3-D space and say ‘fourteen’ to yourself very loudly. Everyone does it." I'm happy with this to visualize high-dimensional spaces and even countably-infinite-dimensional spaces. Vectors are just points in this space.

Are there tricks to visualize uncountably-infinite-dimensional spaces? Since it's uncountable, I don't know how to enumerate axes so that I can just picture a few of them. e.g. how do I think about boundedness, completeness, compactness of subsets of continuous functions (without relying on algebraic manipulations alone)?

dkv
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  • Do you have an example in mind? – Lubin Nov 01 '19 at 21:48
  • Specifically, I started thinking about this in the context of proving the Arzela-Ascoli theorem and whether there's a good way to think about sequences of functions that's somehow analogous to the way we think about sequences of vectors (which are simply points in $R^n$, which is easy to visualize by imagining n=3 and pretending it's arbitrarily large). – dkv Nov 11 '19 at 22:08
  • In a case where you’re working with spaces of functions, I don’t believe that visualization of that sort is particularly useful — but that’s not my field at all, so very likely my opinion should not sway you. – Lubin Nov 12 '19 at 03:03

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An element of an uncountably infinite-dimensional vector space can be visualized by arranging its components with respect to the appropriate basis, numbered by real numbers, along the real line. E.g., a plot of some function defined on the set $R^1$ (say, sine, exponent, etc., and not necessarily continuous) will be such a visualization.

Michael B
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