Jtdylktuy

$l^p$ appears frequently in undergrad real analysis courses, I wonder if there is any strong connection between $l^p$ and $L^p$ space? (Other than they look similar)

I give one definition of $l^p$ I've seen:

$begin{equation} ||s||_p=begin{cases}
sup_{ninmathbb{N}}left(sum_{i=1}^{n}|s_i|^pright)^{1/p},if
~1leq p<infty\
sup_{ninmathbb{N}}|s_n|, if~ p=infty
end{cases}end{equation}$

$l^p$ denotes the space of real sequences s with $||s||_p< infty$

Yes. The connection is, that both are the same kind of construction, but over different measure spaces.

The standard definition of $l^p$ spaces is: $l^p = {x: mathbb N to mathbb R|quad ||x||_p<infty}$ where
begin{align*}
||x||_p = left(sum_{ninmathbb N} |x(n)|^pright)^{1/p}
end{align*}

The standard definition of $L^p$ spaces is ${f: X to mathbb R |quad ||f||_p < infty}$ where $X$ is some compact subset of $mathbb R$ or sometimes even $mathbb R$ and
begin{align*}
||f||_p =left( int _X |f|^p, dlambda right)^{1/p}
end{align*}
integration with respect to the Lebesgue measure on the real line. Look how similar they are. The connection is: Let $(mathbb N, mathcal P(mathbb N), mu)$ be a measure space, where $mu$ is the counting measure, i.e. $mu({n})=1$ for all $nin mathbb N$ and $mu$ $sigma$-additive. Then
begin{align*}
int_{mathbb N} |x|^p, dmu = sum_{ninmathbb N} |x(n)|^p
end{align*}
where $x: mathbb N to mathbb R$ is some measurable function, i.e. a sequence. So they really are nearly the same thing and many measure theoretic results hold for both.

$L^p$ and $ell^p$ spaces both come from the same definition in measure theory. Given a measure space $(Omega,Sigma,mu)$ you can define $L^p(Omega,Sigma,mu)$ as the set of all $mu$-measurable functions defined on $Omega$ such that
$$
int_Omega|f|^pdmu<+infty.
$$
$L^p$ is simply $L^p(mathbb{R},mathcal{B},Leb)$, where $mathcal{B}$ is the Borel $sigma$-algebra and $Leb$ is the Lebesgue measure. On the other hand, $ell^p=L^p(mathbb{N},mathcal{P}(mathbb{N}),c)$ where $mathcal{P}(mathbb{N})$ is the $sigma$-algebra of the parts of $mathbb{N}$ and $c$ is the counting measure.