Talk:Schauder basis

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[edit] why countable?

Why does this article require Schauder bases to be countable? This probably makes it impossible for a nonseparable space to have a Schauder basis, whereas for example, \ell^p(X) has a natural basis with the same cardinality as X.

OK, actually, I think I know the reason: if X is uncountable, then modifications must be made to the definition of infinite sum. Although it is true that the space of finite linear combinations of basis elements is dense in \ell^p(X), the limit points cannot all be limits of sequences of finite partial sums. Rather, we would need to define summation of uncountably many terms as the limit of the uncountable net of finite partial sums. This in itself doesn't really entail a modification of the definition though. We would just have to remove the word "sequence" from the article. You're still taking the limit over all partial sums, you just can't use the word "sequence" to describe such a limit.

So maybe the only reason that Schauder doesn't allow nonseparable spaces in his definition is that he thinks that people who deal with nonseparable spaces are miscreants and deviants? -lethe talk + 19:11, 14 April 2006 (UTC)

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