Vector Spaces: Real Numbers Over Rational Numbers

[arunkp]

Please tell me one of the bases for the infinite dimenional vector space - R (the set of all real numbers) over Q (the set of all rational numbers). The vector addition, field addition and multiplication carry the usual meaning.

 

[Dick]

Why do you think there is one you can describe constructively?

 

[HallsofIvy]

It's not JUST "infinite dimensional". Since the set of real numbers is uncountable, while the set of rational numbers is countable, any basis for the real numbers, as a vector space over the rational numbers, would have to be uncountable- so it is impossible to list them.

Theoretically, you could set of a function, say over [0, 1], such that f(x) for each x gives a "basis" number. If you figure out how to do that, please let me know!