Definition:Lp Space/Vector Space

Definition

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

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

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

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

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

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

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

Also see

• $L^p$ Space forms Vector Space