Can an Uncountable Sum Ever Be Finite-Valued? | Why Measure Infinity?

Traditional infinite sums deal with only COUNTABLY infinitely many terms. But is it ever possible to add up UNCOUNTABLY many terms and get a finite sum? And if so, can it give us a way to extend the dot product from finite-dimensional vectors to functions?

=Chapters=
0:00 - Intro
1:23 - Functions as vectors
3:21 - Uncountable sums
6:45 - Analyzing an uncountable sum
10:52 - Resolution


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A few sidenotes on the video:

► What I've been calling a "dot product" on functions and sequences is known more formally as an "inner product". I believe the term "dot product" is usually reserved for dealing with traditional finite-dimensional vectors.
► I described the "components" of a function as coming from each real number input you can plug in, but that was mainly to supply a hypothetical train of thought that would motivate the inquiry that followed. I think those who work in Functional Analysis usually think of function "components" in a somewhat different way (e.g. a Fourier Series decomposition).


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The animations in this video were mostly made with a homemade Python library called "Morpho".
If you want to play with it, you can find it here:
https://github.com/morpho-matters/mor...