The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$

Suppose first $1\le p<\mathrm{\infty }$ and $f-g\in {\mathcal{N}}_p(\mu )$. Then $\mathrm{\parallel }f-g{\mathrm{\parallel }}_p=0$. Applying Minkowski twice yields [L2, L3] $\mathrm{\parallel }f{\mathrm{\parallel }}_p\le \mathrm{\parallel }g{\mathrm{\parallel }}_p+\mathrm{\parallel }f-g{\mathrm{\parallel }}_p=\mathrm{\parallel }g{\mathrm{\parallel }}_p,\mathrm{\parallel }g{\mathrm{\parallel }}_p\le \mathrm{\parallel }f{\mathrm{\parallel }}_p+\mathrm{\parallel }f-g{\mathrm{\parallel }}_p=\mathrm{\parallel }f{\mathrm{\parallel }}_p,$ so $\mathrm{\parallel }f{\mathrm{\parallel }}_p=\mathrm{\parallel }g{\mathrm{\parallel }}_p$. Thus $\mathrm{\parallel }[f]{\mathrm{\parallel }}_p:=\mathrm{\parallel }f{\mathrm{\parallel }}_p$ is well defined.

For $p=\mathrm{\infty }$, if $f-g\in {\mathcal{N}}_{\mathrm{\infty }}(\mu )$ then $f=g$ almost everywhere. Any essential bound for $f$ is therefore an essential bound for $g$ and conversely, so $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus $\mathrm{\parallel }[f]{\mathrm{\parallel }}_{\mathrm{\infty }}:=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ is well defined.

For separation, $\mathrm{\parallel }[f]{\mathrm{\parallel }}_p=0$ means $\mathrm{\parallel }f{\mathrm{\parallel }}_p=0$, and then [L2] gives $f\in {\mathcal{N}}_p(\mu )$ or ${\mathcal{N}}_{\mathrm{\infty }}(\mu )$. Hence $[f]=[0]$. The converse is immediate because the zero representative has norm $0$.

By [L1] and [L5], each quotient $L^p(\mu )$ is already a real vector space. The representative functionals are homogeneous, and [L3] supplies the triangle inequality on representatives; steps 1.1 and 1.2 show that these formulas depend only on the class, so homogeneity and triangle inequality descend to the quotient.

Steps 2.1 and 1.3 verify the three norm axioms named in [L6]. Therefore $(L^p(\mu ),\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p)$ is a normed space for every $1\le p\le \mathrm{\infty }$.