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Can anyone give me a counterexample?

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    $\begingroup$ This seems like an example of the level of questions that deserve to be answered and then closed. $\endgroup$
    – user5810
    Commented Sep 16, 2010 at 6:58
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    $\begingroup$ $L^\infty$ is normally not separable. (Hint: Take the integers with counting measure.) $\endgroup$
    – J.C. Ottem
    Commented Sep 16, 2010 at 8:06
  • $\begingroup$ Is there any reason for this question to remain open? It seems to be undergraduate level. $\endgroup$
    – S. Carnahan ♦
    Commented Sep 16, 2010 at 8:56
  • $\begingroup$ You might find more explanation on math.stackexchange.com $\endgroup$
    – S. Carnahan ♦
    Commented Sep 16, 2010 at 8:58

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How about an uncountable discrete measure space in which each point has measure one.

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