# Definition of $\ell_\infty$

What does the $\ell_\infty$ space stand for?

Thank you.

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The cursive $\ell$ is \ell. $\ell_\infty$ is the space of all bounded (real or complex) sequences. – Zhen Lin Nov 9 '11 at 21:55
Thanks, @ZhenLin! I remember seeing similar symbols like $\ell_0, \ell_1$ etc. How are they related? – max Nov 9 '11 at 21:58
Did you read the Wikipedia page on $L^p$ spaces? The $\ell_p$ space is just a particular case of $L^p$ space, where the measure is the counting measure. – Asaf Karagila Nov 9 '11 at 22:00
@max: It is also important to add, it's fine to ask for a detailed definition and so on. However it is best to include what you already know, or tried to read which is related to the topic (functional analysis, measure theory, etc.) so the answers can be given to your level. If you just ask this like that, an answer could be a single line giving a measure-theoretic definition which may very well go over your head or an extremely detailed explanation on $\ell_p$ spaces which you already know. – Asaf Karagila Nov 9 '11 at 22:09
@tomcuchta: That is not generally true. E.g., consider a nonzero constant sequence. But it is true for elements of $\bigcup\limits_{p\in [1,\infty)}\ell_p$. – Jonas Meyer Nov 9 '11 at 23:05

## 1 Answer

A web search for "l infinity" led to a Wikipedia page for $L^p$ spaces, which includes a section on $\ell^p$ spaces, which is an alternative notation for $\ell_p$ spaces. As stated there, $\ell^\infty$ is the space of bounded sequences. So $\ell_\infty$ is the same thing with different notation.

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