Definition:Lebesgue Space
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Definition
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:
- $\displaystyle \sum_n \left|{x_n}\right|^p < \infty$
The $L^p$ spaces are function spaces defined using natural generalizations of p-norms for finite-dimensional vector spaces.
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Source of Name
This entry was named for Henri Léon Lebesgue.
However, according to Bourbaki's Topological Vector Spaces (1987) they were first introduced by Frigyes Riesz in 1910.
