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.

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

2 Comments »

[…] We can actually extend what we’ve been doing with Hölder’s inequality and Minkowski’s inequality a little further. Given a metric space , we’ve already discussed the idea of an […]

Pingback by The Supremum Metric « The Unapologetic Mathematician | August 30, 2010 | Reply
[…] finite , Minkowski’s inequality allows us to conclude […]

Pingback by Some Banach Spaces « The Unapologetic Mathematician | August 31, 2010 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

RSS Feeds

RSS - Posts
RSS - Comments
Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
Subjects

Subjects
Archives

August 2010

M T W T F S S

1

2 3 4 5 6 7 8

9 10 11 12 13 14 15

16 17 18 19 20 21 22

23 24 25 26 27 28 29

30 31

« Jul Sep »
Search for:

The Unapologetic Mathematician

Mathematics for the interested outsider