Definition:Lp Space/Vector Space

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$. 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 $\map {L^p} {X, \Sigma, \mu}$ be the $L^p$ space of $\struct {X, \Sigma, \mu}$. Let $+$ denote pointwise addition on $\map {\mathcal M} {X, \Sigma, \R}/\sim$.

Let $\cdot$ be pointwise scalar multiplication on $\map {\mathcal M} {X, \Sigma, \R}/\sim$.

Then we define the vector space $\map {L^p} {X, \Sigma, \mu}$ as:


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

Also see

 * Lp Space forms Vector Space