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Math Help - Cardinality of the continuum

  1. #1
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    Cardinality of the continuum

    Hello,
    I need help with the following problem:

    Prove that Card(l^p) = c (cardinality of the continuum), p >= 1.

    I am not quite sure how to start the proof. I'd appreciate anyone's help. Thank you.
    Last edited by anlys; September 21st 2010 at 04:02 PM.
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  2. #2
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    unable to understand

    I would like to help you but I am unsure what your notation means. You are trying to prove that the cardinality of some set I to the pth power is equal to c? I am sorry if this is a dumb question, this is my first post onto this website.
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  3. #3
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    Hi, that's okay. Actually, that is not an I, it's a lower case for L. The notation l^p refers to the metric space with || x ||_p = (summation |x_i|^p from i = 1 to infinity)^(1/p). That's all I was given in the problem.
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  4. #4
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    I think the additional structure on \ell^p(\mathbb{N}) is a red herring. All you need is that \ell^p\subset\mathbb{R}^\mathbb{N}, and that \lvert\mathbb{R}\rvert=\lvert\{0,1\}^\mathbb{N}\rv  ert.

    Then, \lvert\mathbb{R}^\mathbb{N}\rvert=\lvert(\{0,1\}^\  mathbb{N})^\mathbb{N}\rvert=\lvert\{0,1\}^{\mathbb  {N}\times\mathbb{N}}\rvert=\lvert\{0,1\}^\mathbb{N  }\rvert=\lvert\mathbb{R}\rvert.
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