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l-infinity is not separable

Denote by \ell^\infty the space of all bounded complex(real) sequences. Prove that this space is not separable.

Proof: Consider the set of all sequences made of elements 0,1. These sequences are all in \ell^\infty. If two such sequences are distinct, the distance between them is 1. Moreover, the cardinal of this set of sequences is equal to the cardinal of (0,1), which means the set is uncountable. The balls of radius 1/2 centered in these points of \ell^\infty do not intersect. If there would be a countable dense set, this set would need to have an element in every open ball, which is a contradiction, because we would have a injection from an uncountable set to a countable set.

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  1. math
    April 24, 2013 at 6:45 am

    why the cardinal of this set of sequences is equal to the cardinal of (0,1)?

    • April 24, 2013 at 11:49 am

      The cardinal of the set \{f: \Bbb{N} \to \{0,1\}\} is 2^{\aleph_0}, i.e. the cardinal of the power set \mathcal{P}(N) (the set of parts of \Bbb{N}). In fact you can construct a bijection between the two: to each function f: \Bbb{N} \to \{0,1\} you can associate the set X_f=\{ n \in \Bbb{N} : f(n)=1\}, and this is a bijection.

      To see that the power set of \Bbb{N} has, in fact, the same cardinal number as \Bbb{R} you can take a look at the corresponding Wikipedia page.

      If you still can’t find a proof you can understand, let me know and I’ll post one.

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