Vector space (no topology) basis

[mathman]

Which vector space do you have in mind, exactly? As long as it's non-zero, there is a basis.

Vector space elements are infinite sequences of zeros and ones, with arithmetic mod 2 for the scalars..

 

[black hole 123]

but i do not know at all how to write down a vector basis for the direct product ("hamel basis"), and presumably no one else does either. so this is a nice explicit example of something whose existence is guaranteed by zorn's lemma, but apparently no one has ever seen or explicitly described one.

yes I am pretty sure there's a theorem saying existence of hamel basis is equivalent to AC so if someone managed to write down explicit basis then that can only mean trouble...

this is like the issue with ultrafilters and well ordering of reals and many other things.

 

[WWGD]

Vector space elements are infinite sequences of zeros and ones, with arithmetic mod 2 for the scalars..

Then your scalars are essentially just 0 and 1 then, I guess.

 

[mathman]

Then your scalars are essentially just 0 and 1 then, I guess.

Yes. Scalar arithmetic is mod 2.