One way to understand $l^1(X)$ for a set $X$ with counting measure is that $l^1(-): Set \to Ban$ provides a left adjoint to the functor $\hom(k, -): Ban \to Set$. Here $k$ is the ground field and $Ban$ denotes the category of Banach spaces and linear maps $T: X \to Y$ with $\|T\| \leq 1$. Another way of saying this is that $l^1(X)$ is a coproduct of an $X$-indexed collection of copies of the ground field $k$. Similarly, $l^\infty(X)$ is a product of an $X$-indexed collection of copies of $k$.

Tom Leinster has given a neat description of $L^1[0, 1]$ in terms of universal properties: it is initial among Banach spaces $X$ equipped with maps $u: k \to X$ and $\xi: X \oplus X \to X$ such that $\xi(u, u) = u$. Details can be found here.