The set of real functions has the structure of a vector space, and all vector spaces have basis. What is the basis of this space then? My first thought was maybe using taylor series but still there are a lot of functions that can not be expressed through taylor series and a linear combination can not have infinite elements (I think). Is there a known basis to this vector space? Is there a way to find it?
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3Not even the vector space of continuous functions has a 'nice' basis, see this. One needs to use the axiom of choice to find a basis. – Julio Puerta Mar 30 '24 at 14:17
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"all vector spaces have basis".... says who? – EmmaK Mar 30 '24 at 14:17
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1@EmmaK Most Linear Algebra books that assume the the Axiom of Choice. – jjagmath Mar 30 '24 at 14:31