Definition:Lp Space/Normed Vector Space

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

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

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

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

Let $+$ denote pointwise addition on $\map {\mathcal M} {X, \Sigma, \R}/\sim$.

Let $\cdot$ be pointwise scalar multiplication on $\map {\mathcal M} {X, \Sigma, \R}/\sim$. Let $\norm \cdot_p$ be the $L^p$ norm on $\map {L^p} {X, \Sigma, \mu}$.

Then we define the normed vector space $\struct {\map {L^p} {X, \Sigma, \mu}, \norm \cdot_p}$ by:


 * $\struct {\map {L^p} {X, \Sigma, \mu}, \norm \cdot_p} = \struct {\struct {\map {L^p} {X, \Sigma, \mu}, +, \cdot}_\R, \norm \cdot_p}$

Also see

 * $L^p$ Space forms Normed Vector Space
 * $L^p$ Space is Banach Space