# Definition:P-Seminorm

## Definition

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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

Let $\mathcal{L}^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$.

The $p$-seminorm on $\mathcal{L}^p \left({\mu}\right)$ is the mapping $\left\Vert{\cdot}\right\Vert_p : \mathcal{L}^p \left({\mu}\right) \to \R_{\ge 0}$ defined by:

$\displaystyle \forall f \in \mathcal{L}^p \left({\mu}\right): \left\Vert{f}\right\Vert_p := \left({\int \left\vert{f}\right\vert^p \, \mathrm d \mu}\right)^{1/p}$

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