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 $\map {\mathcal M} {X, \Sigma, \R}/\sim_\mu$ be the space of real-valued measurable functions identified by $\mu$-A.E. equality.

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

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

Let $\norm \cdot_p$ be the $L^p$ norm on $\map {L^p} {X, \Sigma, \mu}$.

From $L^p$ norm is norm, $\norm \cdot_p$ is a norm on $\map {L^p} {X, \Sigma, \mu}$.

So, we can 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 is Banach Space