sudgylacmoe | The Geometric Product Is Not Continuous (In Infinite Dimensions) @sudgylacmoe | Uploaded March 2024 | Updated October 2024, 2 hours ago.
Here's a surprising fact that I discovered a little while ago: in an infinite-dimensional space, the geometric product is not continuous! While infinite-dimensional geometric algebra is perfectly fine and usable, this suggests that infinite-dimensional geometric calculus is practically useless. Interestingly, I came across this fact back when I was trying to prove that the geometric product IS continuous. I kept running into roadblocks, and eventually started wondering if it actually isn't continuous. After running a bit of code, I found that the numerical evidence suggested that it's not, and soon afterwards I found a comment online about something similar (math.stackexchange.com/questions/816092/infinite-dimensional-clifford-algebras#comment6841432_818115), and I adapted that argument into the one shown in the short.

You can find more counterexamples in geometric algebra in this document: drive.google.com/file/d/1BMnv9aZlDcsh4AnivVt7ZIGqmdTcp8tn/view

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