Lp spaces · Definition

Minkowski inequality

measure theory and integrationlp spaces

It is the norm version of the triangle inequality. Together with Hölder's inequality, it lets us prove that \(L^p\) is a normed space.

Definition

Inequality stating that the \(L^p\) norm of a sum of functions is at most the sum of their \(L^p\) norms, for \(p \ge 1\).

\[\|f+g\|_p \le \|f\|_p + \|g\|_p\]

For \(1 \le p < \infty\) and measurable \(f,g\) with finite norms, \(\|f+g\|_p \le \|f\|_p + \|g\|_p\) holds. For \(p>1\) the proof combines Hölder's inequality with the conjugate exponent \(q\); equality occurs when one function is a positive multiple of the other.

With \(f(x)=x\), \(g(x)=1\) on \([0,1]\) and \(p=2\): \(\|f\|_2=1/\sqrt{3}\), \(\|g\|_2=1\), and \(\|f+g\|_2=\sqrt{7/3}\approx 1.53\), which is less than \(\approx 1.58\).

Typical error: assuming the norm of a sum equals the sum of the norms. With \(f=1\) and \(g=-1\) on \([0,1]\), \(\|f+g\|_p=0\), while the sum of norms is \(2\).

Think about it

Why is having the same norm not enough for equality in Minkowski's inequality?

Equal norms only fix the size, not the direction. With \(f=\mathbf 1_{[0,1]}\), \(g=\mathbf 1_{[1,2]}\) and \(p=2\), both norms equal 1, but \(\|f+g\|_2=\sqrt{2}<2\).

Exercises 10

  1. What is the Minkowski inequality for \(p \ge 1\)?
  2. For which values of \(p\) does the Minkowski inequality hold?
  3. Which basic real-number inequality is the Minkowski inequality related to?
  4. If \(p = 1\), what does the Minkowski inequality state?
  5. When does equality hold in the Minkowski inequality for \(1<p<\infty\)?
  6. With \(f(x)=x\) on \([0,1]\) and \(p=2\), what is \(\|f\|_2\)?
  7. With \(f(x)=x\), \(g(x)=1\) on \([0,1]\) and \(p=2\), what is \(\|f+g\|_2\)?
  8. For \(f(x)=x\) and \(g(x)=1\) on \([0,1]\) with \(p=2\), does the Minkowski inequality hold?
  9. With \(p=1/2\), \(f=\mathbf 1_{[0,1]}\) and \(g=\mathbf 1_{[1,2]}\) on \([0,2]\), what is \(\|f+g\|_{1/2}\)?
  10. For \(0<p<1\), which norm property fails for \(\|\cdot\|_p\)?

References

  • Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley, 1999.
  • H. L. Royden, P. M. Fitzpatrick, Real Analysis, 4th edition, Pearson, 2010.