Complex Holder, Minkowski, and the quotient norm

By F1 the real functions $\mathrm{\mid }f\mathrm{\mid },\mathrm{\mid }g\mathrm{\mid }$ are measurable with finite respective norms. Applying F3 to these functions and using $\mathrm{\mid }fg\mathrm{\mid }=\mathrm{\mid }f\mathrm{\mid }\mathrm{\mid }g\mathrm{\mid }$ gives $\int \mathrm{\mid }fg\mathrm{\mid }\le N_p(f)N_{p^{\mathrm{\prime }}}(g)<\mathrm{\infty }$, for $1<p<\mathrm{\infty }$ and also $(p,p^{\mathrm{\prime }})=(1,\mathrm{\infty }),(\mathrm{\infty },1)$. Thus $fg$ is complex integrable and F5 gives the asserted integral bound.

For finite $p$, F2 and F8 give $N_p(f+g)\le N_p(\mathrm{\mid }f\mathrm{\mid }+\mathrm{\mid }g\mathrm{\mid })$. Real Minkowski on the two real nonnegative functions gives $N_p(f+g)\le N_p(f)+N_p(g)$, so addition preserves the finite-functional class.

For $p=\mathrm{\infty }$ put $a=N_{\mathrm{\infty }}(f)$ and $b=N_{\mathrm{\infty }}(g)$. Every $a+\eta$ and $b+\eta$ with $\eta >0$ is an essential bound: an essential bound smaller than it exists by the infimum property. Off the union of two null sets, $\mathrm{\mid }f+g\mathrm{\mid }\le a+b+2\eta$. Taking infima and then $\eta \downarrow 0$ gives $N_{\mathrm{\infty }}(f+g)\le a+b$.

If $f=f_1$ a.e. and $g=g_1$ a.e., then $\alpha f+\beta g=\alpha f_1+\beta g_1$ off the union of their measurable disagreement sets. For finite $p$, $\mathrm{\mid }f{\mathrm{\mid }}^p=\mathrm{\mid }{f}_1{\mathrm{\mid }}^p$ a.e., so F7 makes their integrals equal. At infinity the sets of essential bounds agree. Thus both operations and $N_p$ descend. Pointwise complex vector identities descend as well; scalar closure and $N_p(cf)=\mathrm{\mid }c\mathrm{\mid }{N}_p(f)$ follow from F2 and F8 for finite $p$, and scaling essential bounds for infinity. For $c=0$ this equality is immediate without dividing by $c$.

For finite $p$, $N_p(f)=0$ iff $\mathrm{\mid }f{\mathrm{\mid }}^p=0$ a.e. by F6, iff $f=0$ a.e. by F2. For infinity, if $N_{\mathrm{\infty }}(f)=0$, each measurable set $E_m=\{\mathrm{\mid }f\mathrm{\mid }>1\mathrm{/}m\}$ is null by F9. Since $\{f\mathrm{\ne }0\}={\bigcup }_{m\ge 1}{E}_m$, F10 gives $f=0$ a.e. The converse follows since zero is then an essential bound. This proves positive definiteness, including zero measure spaces.

Finally $\mathrm{\mid }\bar{f}\mathrm{\mid }=\mathrm{\mid }f\mathrm{\mid }$, and pointwise $\mathrm{\mid }\mathrm{Re}f\mathrm{\mid },\mathrm{\mid }\mathrm{Im}f\mathrm{\mid }\le \mathrm{\mid }f\mathrm{\mid }\le \mathrm{\mid }\mathrm{Re}f\mathrm{\mid }+\mathrm{\mid }\mathrm{Im}f\mathrm{\mid }$. Monotonicity of finite integrals or of essential bounds gives the two lower component bounds; the triangle inequality applied to $f=\mathrm{Re}f+i\mathrm{Im}f$ gives the upper bound. Together with homogeneity, definiteness and the descended operations, this proves all norm assertions.