Definition:P-Seminorm

Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $\map {\LL^p} \mu$ be Lebesgue $p$-space for $\mu$.

The $p$-seminorm on $\map {\LL^p} \mu$ is the mapping $\norm \cdot_p : \map {\LL^p} \mu \to \R_{\ge 0}$ defined by:

$\ds \forall f \in \map {\LL^p} \mu: \norm f_p := \paren {\int \size f^p \rd \mu}^{1 / p}$

That the $p$-seminorm is in fact a seminorm is proved on $p$-Seminorm is Seminorm.

Also see

• $p$-Seminorm is Seminorm

• $p$-Norm, a closely related norm.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.5 \ \text{(ii)}$