l-infinity is not separable

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l-infinity is not separable

January 8, 2011 beni22sof Leave a comment Go to comments

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.

Categories: Functional Analysis Tags: separable

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math

April 24, 2013 at 6:45 am

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why the cardinal of this set of sequences is equal to the cardinal of (0,1)?
- beni22sof
  
  April 24, 2013 at 11:49 am
  
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  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.
- MathGeek
  
  September 21, 2020 at 6:16 pm
  
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  Because each number in (0,1) we can write it as a binary representation.
MathGeek

September 21, 2020 at 6:15 pm

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Isn’t this proof considered as example ?
We only considered the sequences of 1s and 0s
What about the rest?
I’m a little confused can you please explain.
Thank you.
- beni22sof
  
  September 21, 2020 at 11:10 pm
  
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  I don’t understand your question. The sequences of 1s and 0s are used to construct an uncountable family of open sets which do not intersect (the balls of radius 1/2 around each one of them).
  
  A space is called “separable” if it has a countable dense subset. Recall that a dense subset must have an element in every open ball. Therefore, if l-infinity were separable, there should be an element from this countable set in every one of the uncountable family of open balls described before.
  
  This is impossible, since there cannot be a surjective map from a countable set onto an uncountable one.