Definition:Lp Norm

Definition

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

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

We define the $L^p$ norm by:

$\ds \norm {\eqclass f \sim}_p = {\norm f}_p = \paren{ \int \size f^p \rd \mu }^{1/p}$

for each $\eqclass f \sim \in \map {L^p} {X, \Sigma, \mu}$, where $\norm \cdot_p$ is the $p$-seminorm.

$L^\infty$ norm

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

We define the $L^\infty$ norm by:

$\norm {\eqclass f \sim}_\infty = \norm f_\infty$

for each $\eqclass f \sim \in \map {L^\infty} {X, \Sigma, \mu}$, where $\norm \cdot_\infty$ is the supremum seminorm.

Also see

• $L^p$ Norm is Well-Defined
• $L^p$ Norm is Norm