# The Unapologetic Mathematician

## Minkowski’s Inequality

We continue our project to show that the $L^p$ spaces are actually Banach spaces with Minkowski’s inequality. This will allow us to conclude that $L^p$ is a normed vector space. It states that if $f$ and $g$ are both in $L^p$, then their sum $f+g$ is in $L^p$, and we have the inequality

$\displaystyle\lVert f+g\rVert_p\leq\lVert f\rVert_p+\lVert g\rVert_p$

We start by considering Hölder’s inequality in a toy space I’ll whip up right now. Take two isolated points, and let each one have measure $1$; the whole space of both points has measure $2$. A function is just an assignment of a pair of real values $(a_1,a_2)$, and integration just means adding them together. Hölder’s inequality for this space tells us that

$\displaystyle\lvert a_1b_1+a_2b_2\rvert\leq\left(\lvert a_1\rvert^p+\lvert a_2\rvert^p\right)^\frac{1}{p}\left(\lvert b_1\rvert^q+\lvert b_2\rvert^q\right)^\frac{1}{q}$

where $p$ and $q$ are Hölder-conjugate to each other. We can set $a_1=\lvert f\rvert^p$, $a_2=\lvert g\rvert^p$, and $b_1=b_2=\lvert f+g\rvert^{p-1}$ and use this inequality to find

\displaystyle\begin{aligned}\lvert f+g\rvert^p&=\lvert f+g\rvert\,\lvert f+g\rvert^{p-1}\\&\leq(\lvert f\rvert+\lvert g\rvert)\lvert f+g\rvert^{p-1}\\&=\lvert f\rvert\,\lvert f+g\rvert^{p-1}+\lvert g\rvert\,\lvert f+g\rvert^{p-1}\\&\leq\left(\lvert f\rvert^p+\lvert g\rvert^p\right)^\frac{1}{p}\left(2\lvert f+g\rvert^{q(p-1)}\right)^\frac{1}{q}\\&\leq\left(\lvert f\rvert^p+\lvert g\rvert^p\right)^\frac{1}{p}2^\frac{1}{q}\lvert f+g\rvert^{p-1}\end{aligned}

Dividing out $\lvert f+g\rvert^{p-1}$ and raising both sides to the $p$th power, we conclude that $\lvert f+g\rvert^p\leq 2^\frac{p}{q}\left(\lvert f\rvert^p+\lvert g\rvert^p\right)$. Thus if both $\lvert f\rvert^p$ and $\lvert g\rvert^p$ are integrable, then so is $\lvert f+g\rvert^p$. Thus $f+g$ must be in $L^p$.

Now we calculate

\displaystyle\begin{aligned}\lVert f+g\rVert_p^p&=\int\lvert f+g\rvert^p\,d\mu\\&\leq\int\lvert f\rvert\,\lvert f+g\rvert^{p-1}\,d\mu+\int\lvert g\rvert\,\lvert f+g\rvert^{p-1}\,d\mu\\&\leq\left(\int\lvert f\rvert^p\,d\mu\right)^\frac{1}{p}\left(\int\lvert f+g\rvert^{q(p-1)}\,d\mu\right)^\frac{1}{q}+\left(\int\lvert g\rvert^p\,d\mu\right)^\frac{1}{p}\left(\int\lvert f+g\rvert^{q(p-1)}\,d\mu\right)^\frac{1}{q}\\&\leq\left(\int\lvert f\rvert^p\,d\mu\right)^\frac{1}{p}\left(\left(\int\lvert f+g\rvert^p\,d\mu\right)^\frac{1}{p}\right)^\frac{p}{q}+\left(\int\lvert g\rvert^p\,d\mu\right)^\frac{1}{p}\left(\left(\int\lvert f+g\rvert^p\,d\mu\right)^\frac{1}{p}\right)^\frac{p}{q}\\&=\left(\lVert f\rVert_p+\lVert g\rVert_p\right)\left(\lVert f+g\rVert_p\right)^\frac{p}{q}\end{aligned}

Dividing out by $\left(\lVert f+g\rVert_p\right)^\frac{p}{q}$ we find that

$\displaystyle\lVert f+g\rVert_p=\left(\lVert f+g\rVert_p\right)^\frac{p}{p}=\left(\lVert f+g\rVert_p\right)^{p\left(1-\frac{1}{q}\right)}=\left(\lVert f+g\rVert_p\right)^{p-\frac{p}{q}}\leq\lVert f\rVert_p+\lVert g\rVert_p$

This lets us conclude that $L^2$ is a vector space. But we can also verify the triangle identity now. Indeed, if $f$, $g$, and $h$ are all in $L^p$, then Minkowski’s inequality shows us that

$\displaystyle\rho_p(f,g)=\lVert f-g\rVert_p\leq\lVert f-h\rVert_p+\lVert h-g\rVert_p=\rho_p(f,h)+\rho_p(h,g)$

which is exactly the triangle inequality we want. Thus $\lVert\cdot\rVert_p$ is a norm, and $L^p$ is a normed vector space.