Uses of notation $\ell^p$ and $L^p$?

Question

On the wikipedia page of $L^p$ spaces, it talks about $\ell^p$ spaces and $L^p$ spaces.

It states that $\ell^p$ spaces are sequence spaces, and $L^p$ spaces are function spaces (with $p$-norm smaller than infinity).

However, I sometimes see $\ell^p$ used without explicitly stating that it means a "space of sequences" (with the appropriate norm condition), and am unsure whether it refers to the same concept.

Do the notations $\ell^p$ and $L^p$ as used in mathematical contexts always refer to these two concepts as stated on the wikipedia page, or are there other common uses of the notation?

Answer 1

Yes, the notation $\ell^p$ almost always refers to the space of sequences as stated on Wikipedia.

This is also true for $L^p$, but it most cases $L^p$ appears together with a measure space, e.g. $L^p(S,\mu)$, where $\mu$ is a measure on $S$. In many cases the measure is not mentioned explicitly, e.g. the notation $L^p(\mathbb R^n)$ or $L^p(\Omega)$ for some set $\Omega\subset\mathbb R^n$ automatically assumes the Lebesgue measure. If $\Omega=(a,b)$ is an open interval, then the space is also sometimes written as $L^p(a,b)$.

Answer 2

The notation of course refers for $ℓ^p$ space to the case of sequences space, in particular given a real sequence $\lbrace x_n \rbrace$ for $p\in[1,+\infty)$, let

$$||\lbrace x_n\rbrace||_{ℓ^p}:=\Biggl ( \sum_{n=1}^\infty |x_n|^p \Biggr )^{1/p}.$$

The space $ℓ^p$ is then the space of those sequences which have this norm finite. The usual construction of function spaces $L^p(\mu)$ allows us to state that $ℓ^p$ spaces are particular cases of $L^p(\mu)$ spaces. This holds true whenever the measure $\mu$ is the counting measure over $\mathbb{N}$.