Definition:Lp Space

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.

Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.

Let $\map {\mathcal M} {X, \Sigma}$ be the set of $\Sigma$-measurable functions on $X$.

Let $\sim$ be the almost-everywhere equality equivalence relation on $\map {\mathcal M} {X, \Sigma}$.

We define the $L^p$ space $\map {L^p} {X, \Sigma, \mu}$ as the quotient space:


 * $\map {L^p} {X, \Sigma, \mu} = \map {\mathcal L^p} {X, \Sigma, \mu}/\sim$

$L^\infty$ norm
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Normed Vector Space
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Also see

 * Definition:Integral on L-1 Space