Vector Space

  • Thread starter arunkp
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
Dick
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Why do you think there is one you can describe constructively?
 
  • #3
HallsofIvy
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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!
 

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