Definition:Lebesgue Space

Definition
For a real number $p \ge 1$, $\ell^p$ is the subspace of $\C^\N$ (all complex sequences) consisting of all sequences $\mathbf{x} = \langle{x_n}\rangle$ satisfying:
 * $\displaystyle \sum_n \left\vert{x_n}\right\vert^p < \infty$

The $L^p$ spaces are function spaces defined using natural generalizations of p-norms for finite-dimensional vector spaces.

In a Lebesgue space, products and exponentiation are defined term-wise. That is, $\langle {x_n}\rangle \langle {y_n} \rangle = \langle {x_n y_n} \rangle$ and $\langle {x_n} \rangle^\alpha = \langle {x_n^\alpha} \rangle$.

However, according to Bourbaki's Topological Vector Spaces (1987) they were first introduced by Frigyes Riesz in 1910.