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$L_p$ is often used to describe a norm, or a vector space with that norm (see e.g. wikipedia). Is $\ell_p$ (typically, or canonically) a different notation for the same concept, or is it used to indicate something different?

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$\ell^p$ spaces are particular cases of $\mathbb{L}^p$ spaces. Usually, one uses $\ell^p$ when the underlying space is $\mathbb{Z}$ or $\mathbb{N}$, but I believe I've already seen such things as $\ell^p (\mathbb{Z}^n)$ (everything is done with the counting measure). Since it is a special case, there are a few properties that hold for $\ell^p$ spaces and not for general $\mathbb{L}^p$ spaces, such as $\ell^p \subset \ell^q$ if $p \leq q$. –  D. Thomine Feb 14 '12 at 21:47
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Traditionally, $\ell^p$ is used when the norm involves a summation, while $L^p$ is used when the norm involves an integral. Of course, in modern Lebesgue theory, a summation is a special case of an integral. –  Jim Belk Feb 15 '12 at 0:14

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up vote 9 down vote accepted

$\ell^p$ spaces are a special case of $L^p$ spaces.

If $(X,\mu)$ is a measure space, $L^p(X)$ (or $L^p_{\mathbb{R}}(X)$) is the (Banach) space of all measurable functions $f\colon X\to \mathbb{R}$ such that $$\int_X |f|^p\,d\mu\lt \infty.$$

In the special case in which $X=\mathbb{N}$ and $\mu$ is the counting measure, functions $f\colon\mathbb{N}\to\mathbb{R}$ can be taken to be sequences of elements of $\mathbb{R}$, and the integral is the sum of the terms of the sequence. That is, $L^p(\mathbb{N})$ is the set of sequences $(x_i)$ such that $\sum |x_i|^p\lt\infty$. To denote this special case, which occurs very often, we use $\ell^p$.

(You can replace $\mathbb{R}$ with any normed vector space, replacing the absolute value with the norm.)

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For L p space or l p space, I was wondering when to use subscript and when to use superscript? Why is that? –  Tim Feb 15 '12 at 2:18
    
@Tim: One usually uses superscript, because it is mnemonic that the $p$ is the exponent in the condition, and because it leaves the subscript available to indicate the range of the functions. –  Arturo Magidin Feb 15 '12 at 4:19
    
Shouldn't there be a $()^{1/p}$ in there somewhere (e.g. the Euclidean norm)? –  Joe Feb 15 '12 at 8:54
    
@Joe: The $()^{1/p}$ is what you do in order to calculate the $p$-norm; but $f$ is in $L^p$ if and only if $\int |f|^p d\mu$ is finite. Similarly, the $p$-norm of a sequence in $\ell^p$ is $(\sum |x_i|^p)^{1/p}$, but whether a sequence is in $\ell^p$ or not is determined by looking at $\sum |x_i|^p$. –  Arturo Magidin Feb 15 '12 at 16:06
    
@ArturoMagidin: Thanks! I suspect there is some inconsistency in superscript or subscript in your reply. –  Tim Feb 15 '12 at 18:19

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