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Can we reproduce a basis of a vector space of all sequences?(if exists since by every vector space has a basis, but can we reproduce it)

Ben Grossmann
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Sameer Pande
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    I think you mean "produce" not "reproduce," but the usual term would be "construct." I would guess that the answer is "No," but I don't know how to prove it. (I assume you mean construct it without using the axiom of choice.) – saulspatz Feb 23 '18 at 20:20
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    The general questions is discussed at https://math.stackexchange.com/questions/86762/finding-a-basis-of-an-infinite-dimensional-vector-space but your specific example is not mentioned. – saulspatz Feb 23 '18 at 20:25
  • It's already hard to even think of describing a basis that has to consist of uncountably many members, when the total description has to fit into finitely many words ... – Hagen von Eitzen Feb 23 '18 at 20:25
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    @HagenvonEitzen: Describing a basis with uncountably many members is not necessarily an unsolvable problem; for example a basis for the vector space of functions $\mathbb R\to\mathbb R$ that are non-zero at only finitely many points is certainly uncountable, yet I can easily write it down even inside this comment: It's given by the set of functions that are $1$ at exactly one point, and $0$ otherwise. – celtschk Feb 23 '18 at 20:41

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