On Mon, Aug 27, 2012 at 7:06 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
So the core paradox seems to reside in the following counterintuitive phenomenon, once (whatever) dimension of a vector space etc. becomes infinite. The cardinal K1 such that every vector is expressible as a linear combination of some K1 points, and the cardinal K2 such that every subset of K2+1 points is linearly dependent, are no longer necessarily equal.
This is just because you're stating things in a way that works for finite and not infinite sets. The K1 such that every vector is expressible as a linear combination of some K1 points, and the cardinal K2 such that every set of cardinality > K2 is linearly dependent, are equal. It's just that using K2+1 gives the same result as > K2 when K2 is finite, but not when it's infinite. This is all using vector space, not Hilbert space, definitions, where there is no topology, so no way to add more than finitely many vectors, and linear combination and linear dependence are both defined in terms of finite sums. The theorem also works for Hilbert basis, where you allow countable sums. The cardinal K1 such that every vectorr is expressible as a linear combination of countably many of some K1 points, and the cardinal K2 such that every subset of > K2 points is linearly dependent in the sense that there is a countable set of multiples of elements of the set, not all 0, that sums to 0. If you use finite sums to define a basis, and countable sums to define linear independence, or vice versa, you get different cardinals, but that's to be expected.
Is it obvious that (Hilbert dimension) K1 <= K2 (vector-space dimension) ?
Yes; if every vector is a finite sum of elements of S, it's also a countable sum of elements of S. As far as your later comment goes, I don't know of any useful definition of either basis or linear independence that makes use of sums of more than countably many vectors. I don't know of any notion of summability of uncountable sets that doesn't consider any sum with uncountably nonzero elements to diverge. Andy