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## l-infinity is not separable

Denote by 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 . These sequences are all in . If two such sequences are distinct, the distance between them is . Moreover, the cardinal of this set of sequences is equal to the cardinal of , which means the set is uncountable. The balls of radius centered in these points of 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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Categories: Functional Analysis
separable

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

The cardinal of the set is , i.e. the cardinal of the power set (the set of parts of ). In fact you can construct a bijection between the two: to each function you can associate the set , and this is a bijection.

To see that the power set of has, in fact, the same cardinal number as 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.