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What applications are there of the fact that the real numbers form a vector space over the rational numbers?

Vector spaces over the rational numbers appear to have uses in number theory.

The motivation for this question comes from Dietrich's answer to this question: Applications of uncountability of the real numbers. That question asks why it is useful to know that the real numbers are uncountable. This was motivated by an open problem. The most popular answer to that question mentioned the fact that the real numbers form an infinite-dimensional vector space over the rational numbers. This has prompted me to ask why this fact is again useful.

Martin Sleziak
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wlad
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    The google search "Hamel basis" + pathology should give you more than enough applications. – Dave L. Renfro Dec 11 '19 at 11:18
  • @DaveL.Renfro If you can summarise one of the thousands of paper you refer to below, it would make a good answer – wlad Dec 11 '19 at 11:27
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    @jkabrg: I think the best thing would be to use google scholar with selected search terms (e.g. "Hamel basis", "Cauchy" + "functional equation", etc.) and an area of math or topic you know something about. Maybe include "counterexample" also as a search word. Also, I'm very busy right now with some work that I'm trying to get done in the next 8-10 hours. – Dave L. Renfro Dec 11 '19 at 11:47

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There are nonlinear real functions that satisfy Cauchy's functional equation $f(x+y)=f(x)+f(y)$.

lhf
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Probably the best known application is the construction of a Hamel basis (http://mathworld.wolfram.com/HamelBasis.html ), which allows you to solve the Cauchy functional equation (over the reals): $$f(x+y) = f(x) +f(y) $$ If $f$ is taken to be over the rationals then $f(x)=cx$ is a solution for each $c\in {\mathbb R}$, but if $f$ is taken over the reals then the Hamel basis is used to show that there are infinitely many solutions.

postmortes
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