Relationship between $l^p$ and $L^p$ space? $l^p$ 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:
$\begin{equation} ||s||_p=\begin{cases}
\sup_{n\in\mathbb{N}}\left(\sum_{i=1}^{n}|s_i|^p\right)^{1/p},if     
~1\leq p<\infty\\
\sup_{n\in\mathbb{N}}|s_n|, if~ p=\infty
\end{cases}\end{equation}$ 
$l^p$ denotes the space of real sequences s with  $||s||_p< \infty$
 A: 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_{n\in\mathbb N} |x(n)|^p\right)^{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\, d\lambda \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 $n\in \mathbb N$ and $\mu$ $\sigma$-additive. Then
\begin{align*}
\int_{\mathbb N} |x|^p\, d\mu = \sum_{n\in\mathbb 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.
A: $L^p$ 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
$$
\int_\Omega|f|^pd\mu<+\infty.
$$
$L^p$ 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.
