The Unapologetic Mathematician

Mathematics for the interested outsider

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 pth 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.

August 27, 2010 - Posted by | Analysis, Measure Theory


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