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Prove that a normed space is not Banach

  1. Feb 16, 2012 #1
    Hello everyone,
    I have a problem and cannot solve it. Could you help? Here it is
    We have a normed space and an uncountable Hamel basis of it. Prove that it is not a Banach space.
    Should I use Baire theorem? Any suggestions?
  2. jcsd
  3. Feb 16, 2012 #2
    You're not giving us much information to work with. If you want to show something not Banach, then try to find a Cauchy sequence which does not converge.
    There are other ways to prove this of course. A lot depends on the space in question.
  4. Feb 16, 2012 #3
    You must have a particular space in mind that you didn't specify. An infinite dimensional Banach space necessarily has an uncountable Hamel basis so your result is not general. You need to show that your space is not complete. What is your space?
  5. Feb 16, 2012 #4
    The exercise does not refer to a particular space. It is just a normed space X with an uncountable Hamel basis. A solution I came up with was to make a finite dimension closed subspace and show using baire that it is X, leading to a contadiction.
    If the problem gave us a space for this example which one would that be?
  6. Feb 16, 2012 #5
    Could you quote the exercise exactly as it was given??
  7. Feb 16, 2012 #6


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    But, how could you prove that a closed subspace of an undefined space equals the space itself?
  8. Feb 16, 2012 #7
    Also notice that a finite dimensional normed space is always Banach, so your proof is likely incorrect.
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