Proof of Minkowski's Inequality

Why a Proof Needs More Than the Pointwise Triangle Inequality

Minkowski’s inequality was stated in the previous tutorial: the \(L^p\) norm of a sum is at most the sum of the \(L^p\) norms. The scalar estimate \(|f+g|\leq |f|+|g|\) is an important first step, but for \(1<p<\infty\) it does not by itself give the desired bound after taking the \(p\)th power and integrating. The proof uses Hölder’s inequality to control the resulting products.

We work on an arbitrary measure space \((X,\mathcal{F},\mu)\), with real-valued functions understood up to equality almost everywhere. For \(1\leq p<\infty\), \(\|f\|_p=(\int_X |f|^p\,d\mu)^{1/p}\); for \(p=\infty\), \(\|f\|_\infty\) is the essential supremum. The linear-closure result for \(L^p\) established earlier ensures that \(f+g\in L^p\) whenever \(f,g\in L^p\).

The Endpoint Exponents

At \(p=1\), integrate the pointwise triangle inequality. Since \(f,g\in L^1\), both \(|f|\) and \(|g|\) have finite integrals, and the pointwise bound gives $$ \int_X |f+g|\,d\mu \leq \int_X (|f|+|g|)\,d\mu =\|f\|_1+\|g\|_1. $$ The last equality uses additivity of the nonnegative integral. This proves Minkowski’s inequality for \(p=1\).

For \(p=\infty\), the least almost-everywhere bound property of the essential supremum gives \(|f|\leq\|f\|_\infty\) and \(|g|\leq\|g\|_\infty\) almost everywhere. Outside the union of the two exceptional null sets, the scalar triangle inequality therefore gives $$ |f+g|\leq |f|+|g|\leq\|f\|_\infty+\|g\|_\infty. $$ Thus the right side is an almost-everywhere bound for \(|f+g|\), so \(\|f+g\|_\infty\leq\|f\|_\infty+\|g\|_\infty\).

The Proof for Intermediate Exponents

Now suppose \(1<p<\infty\), and let \(q=p/(p-1)\), so \(1/p+1/q=1\). Put \(h=f+g\). The essential calculation is that the factor \(|h|^{p-1}\) has exactly the integrability needed to pair with \(f\) and \(g\) in Hölder’s inequality.

Proof. Since \((p-1)q=p\), $$ \int_X \bigl||h|^{p-1}\bigr|^q\,d\mu =\int_X |h|^{(p-1)q}\,d\mu =\int_X |h|^p\,d\mu<\infty. $$ Taking the \(q\)th root gives $$ \bigl\||h|^{p-1}\bigr\|_q =\left(\int_X |h|^p\,d\mu\right)^{1/q} =\|h\|_p^{p/q} =\|h\|_p^{p-1}, $$ because \(p/q=p-1\). This proves the lemma.

This argument explains why Hölder’s inequality is the central tool for intermediate exponents. The factor \(|h|^{p-1}\) appears when the \(p\)th power of \(|h|\) is written as \(|h|\,|h|^{p-1}\). Its conjugate exponent norm is precisely \(\|h\|_p^{p-1}\), which cancels after division and leaves the desired first power of \(\|h\|_p\).

When Equality Holds for \(1<p<\infty\)

For real-valued functions and a finite exponent strictly greater than one, equality has a precise form. In proving it, care is needed on the set where \(h=f+g\) vanishes: equality in the first integral estimate alone says nothing there, because its factor \(|h|^{p-1}\) is zero. The Hölder equalities provide the missing information.

Proof. If one function is zero almost everywhere, equality follows directly. If \(g=cf\) almost everywhere for \(c>0\), then \(f+g=(1+c)f\) almost everywhere. Homogeneity gives $$ \|f+g\|_p=(1+c)\|f\|_p=\|f\|_p+\|g\|_p. $$ This proves sufficiency.

For necessity, suppose equality holds and both norms \(\|f\|_p\) and \(\|g\|_p\) are positive. Set \(h=f+g\). Then \(\|h\|_p=\|f\|_p+\|g\|_p>0\). In the proof of Minkowski’s inequality, the first integral is bounded by the sum of two Hölder bounds: $$ \int_X|h|^p\,d\mu \leq \int_X(|f|+|g|)|h|^{p-1}\,d\mu \leq(\|f\|_p+\|g\|_p)\|h\|_p^{p-1}. $$ Under the assumed norm equality, the first and last expressions are equal. Therefore both inequalities in this chain are equalities. Moreover, each of the two Hölder deficits is nonnegative, and their sum is zero; hence each individual Hölder inequality is an equality. By the equality condition in Hölder’s inequality, together with the positive norms, this gives $$ |f|=\frac{\|f\|_p}{\|h\|_p}|h|, \qquad |g|=\frac{\|g\|_p}{\|h\|_p}|h| \quad\text{almost everywhere}. $$ In particular, \(f=g=0\) almost everywhere on \(\{h=0\}\).

The equality in the first integral estimate also means that the nonnegative function $$ (|f|+|g|-|h|)|h|^{p-1} $$ has integral zero. By the Zero Integral Criterion, it is zero almost everywhere. On \(\{h\ne0\}\), the factor \(|h|^{p-1}\) is positive, so \(|f+g|=|f|+|g|\) almost everywhere there. For real numbers, this scalar equality holds exactly when the two numbers have the same sign or at least one is zero. The displayed proportionality relations show that, wherever \(h\ne0\), both \(f\) and \(g\) have the sign of \(h\). Thus, on that set, $$ f=\frac{\|f\|_p}{\|h\|_p}h, \qquad g=\frac{\|g\|_p}{\|h\|_p}h. $$ These equations also hold on \(\{h=0\}\), because \(f=g=0\) there almost everywhere. Consequently \(g=cf\) almost everywhere, where \(c=\|g\|_p/\|f\|_p>0\). This proves necessity and completes the proof.

How to Use the Proof—and a Common Pitfall

The proof separates three tasks. First, pointwise addition gives a scalar estimate. Second, integration turns that estimate into an inequality involving products. Third, Hölder’s inequality controls those products in terms of norms. At the endpoints, the middle step is simpler: direct integration handles \(p=1\), while almost-everywhere bounds handle \(p=\infty\).

A common pitfall is to stop at \(|f+g|\leq|f|+|g|\) and assume that taking \(p\)th powers immediately proves Minkowski’s inequality. For \(p>1\), the resulting expression involves \((|f|+|g|)^p\), and that expression does not yield the required sum of norms by the pointwise triangle inequality alone. The proof above instead pairs \(|f|\) and \(|g|\) with \(|f+g|^{p-1}\), then applies Hölder.