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This is one of those math questions which, now that I have read them, I really NEED to know the answer: does the ℂ-vector space of continuous functions ℝ→ℂ have a basis which is stable under translation? So far, no answer! https://mathoverflow.net/q/392014/17064
Yves de Cornulier pointed out that this is equivalent to asking whether it is a free module over the group algebra of ℝ (the additive group of the reals) over ℂ; it is a torsion-free module, but for non-finite-type modules, going from “torsion-free” to “free” is a big deal.
I posted an answer to a related question by showing that some related, though not entirely natural, vector space of functions f:ℝ→ℂ (those for which {k∈ℤ | f(t+k)≠0} is bounded for every t∈ℝ) DOES NOT admit a translation-stable basis. https://mathoverflow.net/a/392077/17064
Updates on this:
⁃ At the start of this thread I forgot “with compact support” after “continuous functions”: Pierre Colmez answers the erroneous question in 1 tweet: https://twitter.com/ColmezPierre/status/1390363002835189770
⁃ Harry West wrote a very nice answer to the (correct) question: https://mathoverflow.net/a/392147/17064
⁃ At the start of this thread I forgot “with compact support” after “continuous functions”: Pierre Colmez answers the erroneous question in 1 tweet: https://twitter.com/ColmezPierre/status/1390363002835189770
⁃ Harry West wrote a very nice answer to the (correct) question: https://mathoverflow.net/a/392147/17064
⁃ I asked the related (simpler) question of whether ℓ²(ℤ;ℂ) has a vector space basis that is stable under shifts, and the answer is negative like the above: https://mathoverflow.net/a/392267/17064
There is probably a better way to frame all of this.
There is probably a better way to frame all of this.
