Definition:Lebesgue Space

For $$1 < p < \infty$$, $$\ell^p$$ is the subspace of $$\C^\N$$ (all complex sequences) consisting of all sequences $$x = \left({\mathbf{x}_n}\right)$$ satisfying
 * $$\sum_n \left|{x_n}\right|^p < \infty$$

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

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