The following question evolved during a lecture in the Linear Algebra course for M.S. students:
Do we have an explicit basis for the vector space V:= {(x_1, x_2, \ldots, x_n, \ldots ): x_i \in \mathbb{R}, \forall i \geq 1} over the field \mathbb R of reals? Or, has it been proved in literature that one cannot get hold of an explicit basis for this vector space?
A basis of this space is uncountable: The subspace \ell^2 of square summable sequences is an infinite dimensional Hilbert space and so (by the Baire Category Theorem) can't have a countable vector space basis.
ReplyDeleteSince the basis is uncountable I would guess it would be hard to describe!
How about taking the field F_2 with two elements instead of \mathbb{R}?
Here's a link with rather detailed descriptions - focussed also on what "explicit" could mean: http://mathoverflow.net/questions/5303/basis-of-linfinity
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