Notation: $L_p$ vs $\ell_p$

$L_p$ 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?

$\ell^p$ 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.)