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 Seminorm Axiom $\text N 2$: Positive Homogeneity and Seminorm Axiom $\text N 3$: Triangle Inequality.

Proof of Seminorm Axiom $\text N 2$: Positive Homogeneity

Let $f \in \map {\LL^p} {X, \Sigma, \mu}$ and $\lambda \in \R$, we have:

\(\ds \norm {\lambda f}_p\)

\(=\)

\(\ds \paren {\int \size {\lambda f}^p \rd \mu}^{1/p}\)

Definition of P-Seminorm

\(\ds \)

\(=\)

\(\ds \paren {\int \size \lambda^p \size f^p \rd \mu}^{1/p}\)

\(\ds \)

\(=\)

\(\ds \paren {\size \lambda^p \int \size f^p \rd \mu}^{1/p}\)

Integral of Positive Measurable Function is Positive Homogeneous

\(\ds \)

\(=\)

\(\ds \size \lambda \paren {\int \size f^p \rd \mu}^{1/p}\)

\(\ds \)

\(=\)

\(\ds \size \lambda \norm f_p\)

Definition of P-Seminorm

$\Box$

Proof of Seminorm Axiom $\text N 3$: Triangle Inequality

This is Minkowski's Inequality on Lebesgue Space.

$\blacksquare$