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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.2011-11-09
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    Thanks, @ZhenLin! I remember seeing similar symbols like $\ell_0, \ell_1$ etc. How are they related?2011-11-09
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    Did you read [the Wikipedia page on $L^p$ spaces](http://en.wikipedia.org/wiki/Lp_space#lp_spaces)? The $\ell_p$ space is just a particular case of $L^p$ space, where the measure is the counting measure.2011-11-09
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    @max: I don't think I've ever seen $\ell_0$. $\ell_p$ for $1 \le p < \infty$ is the space of sequences with finite $p$-norm; for example, $\ell_1$ is the space of absolutely summable sequences, $\ell_2$ is the space of square-summable sequences, and so on.2011-11-09
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    @max : they are related in that they are all normed spaces of sequences; the difference is that each $l_p$ has a different norm; in a sense, the $l_\infty$ norm is the limit at infinity of the other $l_p$ norms.2011-11-09
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    @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.2011-11-09
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    You can show that norm on $\ell_{\infty}$, $||\cdot||_{\infty}$ equals $\lim_{p \rightarrow \infty} || \cdot ||_p$ where $||\cdot ||_p$ is the norm of $\ell^p$.2011-11-09
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    @tomcuchta: That is not generally true. E.g., consider a nonzero constant sequence. But it is true for elements of $\bigcup\limits_{p\in [1,\infty)}\ell_p$.2011-11-09

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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.