Lebesgue Space

In Functional Analysis, a Lebesgue Space, denoted by $L^p$, where $p>0$, acts of measurable Functions $f$, such that:

$$L^p=\left\{f|{\int }_X|f{|}^{p}d\mu <\infty \right\}$$

• $L^p$ is the Lebesgue Space
  • $L$ stands for Lebesgue, and signals we are using the Lebesgue Integral
  • $p$ is the exponent of the Lebesgue Space and informs its flavor

Space	Space Type	Geometry
$L^p,p\in (0,1)$	F-Space	Non-convex, no norm
$L^p,p\ge 1$	Banach Space	Convex, has norm
$L^p,p=2$	Hilbert Space	Inner Product, Parallelogram Law

• ${\int }_X\dots d\mu$ is a Lebesgue Integral and the environment of the expression
  • $X$ is the Domain in which the functions exist
  • $\mu$ is the measure in $X$, e.g. in 1D space it would be length
• $|f{|}^p$ is the Transformation
• ${\int }_X|f{|}^{p}d\mu <\infty ⟺f\in L^p$