# Vector Space

1. Jun 4, 2008

### 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.

2. Jun 4, 2008

### Dick

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

3. Jun 5, 2008

### HallsofIvy

Staff Emeritus
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!