(draft) Banach algebras and geometric series #318
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Adding dependent pair technique for convergence, and unif -> ptws
zstone1
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@zstone1 why did you close this one? |
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Oh, sorry I should have left a closing comment. With #397 (Arzela-Ascoli), it will dramatically simplify this proof. So while the banach space definitions are still useful, that should be a separate diff. |
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Building on my previous PR that defines the topology of compact convergence, I prove that the geometric series converges compactly in the open unit ball. This is very much a draft, as it does an OK job introducing Banach algebras. I should probably build the whole hierarchy of
NormedRing-> NormedAlgebra -> BanachAlgebra. Too bad Hierarchy Builder can't be applied for this PR.This is an essential lemma in complex analysis and spectral theory, because it'll let contour integrals exchange freely with geometric series.