What are the differences between $l^p$ space and $L^p$ space? I always thought small $l^p$ space, or so-called Banach space, is the space equipped with a norm similar to vector norm as
$$||x||_p = \left( \sum_{i\in\mathbb{N}} |x_i|^p \right)^{\frac{1}{p}}$$
Then $L^p$ space is the space equipped with the $L^p$ norm on an arbitrary function $f$ as
$$||f||_p = \left( \int_S |f|^p \ d\mu \right)^{ \frac{1}{p}}$$
So apparently the difference comes from the integral operator, as that how we measure "how closely" two functions $f$ and $g$ away from each other. Is it fair to say the difference is the integral opertor?

The $\ell^p$ spaces are a special case of the $L^p$ spaces obtained by using the counting measure on the set of natural numbers. If you squint closely at the integral it looks like a sum or indeed as Forever Mozart points out: summation is just integration with the trivial measure on $\mathbb{N}$.
