Category:Definitions/Lp Spaces
Jump to navigation
Jump to search
This category contains definitions related to Lp Spaces.
Related results can be found in Category:Lp Spaces.
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.
Let $\map \MM {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.
Let $\map {\LL^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.
Let $\sim_\mu$ be the restriction of the $\mu$-almost-everywhere equality relation on $\map \MM {X, \Sigma, \R}$ to $\map {\LL^p} {X, \Sigma, \mu}$.
We define the $L^p$ space $\map {L^p} {X, \Sigma, \mu}$ as the quotient set:
\(\ds \map {L^p} {X, \Sigma, \mu}\) | \(=\) | \(\ds \map {\LL^p} {X, \Sigma, \mu} / \sim_\mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\eqclass f {\sim_\mu}: f \in \map {\LL^p} {X, \Sigma, \mu} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\set {g \in \map {\LL^p} {X, \Sigma, \mu} : f = g \, \mu\text{-almost everywhere} }: f \in \map {\LL^p} {X, \Sigma, \mu} }\) |
Pages in category "Definitions/Lp Spaces"
The following 7 pages are in this category, out of 7 total.