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Let {$e_{\alpha}$} be a Hamel basis of $X$, which is a infinite dimensional Banach space, i.e., every vectoe can be represented as $$x=\sum_{finite} f_{\alpha}(x)e_{\alpha}$$ Show that the functionals $f_{\alpha}:X \to K$ are linear. Show that at least one of them is discontinuous.

I really don't know what to do since $f_{\alpha}$ does not need to be the coordinate functionals, does it? And I have a index set, say $\Omega$ which does not need countable so I don't know how to start.

MGF01
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  • It seems to me that by definition, $f_\alpha$ is the $\alpha$th coordinate. I don't know what you mean when you say it does not need to be the coordinate functional. – saulspatz Apr 28 '19 at 13:00
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    You can do better. See this. – David Mitra Apr 28 '19 at 13:05
  • I mean that I didn't know if indeed it is the $\alpha$th coordinate functional. – MGF01 Apr 28 '19 at 13:05
  • David Mitra, I have seen that but there the set of indeces is countable and here it is not. I don't know how to use the argument in your link above. – MGF01 Apr 28 '19 at 14:53
  • @IreneGil The linked question does not assume anything of the sort. In fact, a Hamel basis for an infinite-dimensional Banach space is uncountable or finite. What the question does is assume there are infinitely many coordinate functions that are continuous (perhaps uncountably many), and then take a countable subset of these to form a (countable) sum of vectors that produces a contradiction. – Theo Bendit Apr 28 '19 at 21:28

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