# The Unapologetic Mathematician

## The Supremum Metric

We can actually extend what we’ve been doing with Hölder’s inequality and Minkowski’s inequality a little further. Given a metric space $(X,\mathcal{S},\mu)$, we’ve already discussed the idea of an “essentially bounded” function — one for which there is some real constant $c$ so that $f(x)\leq c$ for almost all $x\in X$. We will write $L^\infty(X)$ for the collection of essentially bounded functions on the measure space. It should be clear that these form a vector space.

We also discussed the “essential supremum” $\text{ess sup}(\lvert f\rvert)$ of an essentially bounded function. We’ll now write this as $\lVert f\rVert_\infty$, suggesting that it’s a norm. And it’s clear that $\lVert cf\rVert_\infty=\lvert c\rvert\lVert f\rVert_\infty$, and that $\lVert f\rVert_\infty=0$ if and only if $f=0$ almost everywhere. Verifying the triangle identity is exactly Minkowski’s inequality.

And, indeed, we know that $\lvert f(x)\rvert\leq\lVert f\rVert_\infty$ and $\lvert g(x)\rvert\leq\lVert g\rVert_\infty$ a.e., so $\lvert f(x)+g(x)\rvert\leq\lvert f(x)\rvert+\lvert g(x)\rvert\leq\lVert f\rVert_\infty+\lVert g\rVert_\infty$ a.e., so whatever the least such essential upper bound is smaller still. That is, $\lVert f+g\rVert_\infty=\lVert f\rVert_\infty+\lVert g\rVert_\infty$.

Now for Hölder’s inequality. For this purpose we consider $\frac{1}{\infty}=0$, and thus $\frac{1}{1}+\frac{1}{\infty}=1$, which means that $1$ and $\infty$ are Hölder-conjugates. Thus our assertion is that if $f$ is integrable and $g$ is essentially bounded, then $fg$ is integrable and $\lVert fg\rVert_1\leq\lVert f\rVert_1\lVert g\rVert_\infty$. Indeed, we know that $\lvert g(x)\rvert\leq\lVert g\rVert_\infty$, and so $\lvert f(x)g(x)\rvert=\lvert f(x)\rvert\lvert g(x)\rvert\leq \lvert f(x)\rvert\lVert g\rVert_\infty$ — both inequalities holding almost everywhere. From this, we conclude that

\displaystyle\begin{aligned}\lVert fg\rVert_1&=\int\lvert fg\rvert\,d\mu\\&\leq\int\lvert f\rvert\lVert g\rVert_\infty\,d\mu\\&=\int\lvert f\rvert\,d\mu\lVert g\rVert_\infty\\&=\lVert f\rVert_1\lVert g\rVert_\infty\end{aligned}

as we asserted. From now on, we’ll allow $p=\infty$ (and $q=1$) whenever we’re talking about a Hölder-conjugate pair or $L^p$-space.