P-Seminorm is Seminorm

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

Let $p \in \hointr 1 \infty$.

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

Let $\norm {\, \cdot \,}_p$ be the $p$-seminorm on $\map {\LL^p} {X, \Sigma, \mu}$.

Then $\norm {\, \cdot \,}_p$ is a seminorm on $\map {\LL^p} {X, \Sigma, \mu}$.

Proof
Let $f \in \map {\LL^p} {X, \Sigma, \mu}$.

From the construction of the integral of a positive $\Sigma$-measurable function, we have:


 * $\ds \paren {\int \size f^p \rd \mu}^{1/p} \ge 0$

so:


 * $\norm f_p \ge 0$

We also have:


 * $\ds \int \size f^p \rd \mu < \infty$

So $\norm {\, \cdot \,}$ maps from $\map {\LL^p} {X, \Sigma, \mu}$ to the non-negative reals.

We now verify $(\text N 2)$ and $(\text N 3)$ in the definition of a seminorm.

Proof of $(\text N 2)$
Let $f \in \map {\LL^p} {X, \Sigma, \mu}$ and $\lambda \in \R$, we have:

Proof of $(\text N 3)$
This is Minkowski's Inequality for Lebesgue Spaces.