Properties of the Lp Norm

From Size and Scaling to Addition

The previous tutorial established that $L^p$ norms are well-defined on almost-everywhere equivalence classes, scale correctly, and respect almost-everywhere order. One central property remains: adding two functions should not produce a norm larger than the sum of their norms. Proving this triangle inequality requires a way to control integrals of products. Hölder’s inequality supplies that estimate, and the Minkowski inequality then proves the triangle property.

The arguments differ slightly at the endpoints. For $p=1$, the triangle inequality for integrals from earlier in this course gives the result directly. For $1<p<\infty$, Hölder’s inequality is the key step. For $p=\infty$, the result follows from almost-everywhere bounds. We treat all three cases so the norm properties hold across the full range $1\leq p\leq\infty$.

Conjugate Exponents and Young’s Inequality

For $1<p<\infty$, the exponent conjugate to $p$ is the number $q$ satisfying $1/p+1/q=1$. Equivalently, $q=p/(p-1)$. The product of a function in $L^p$ and one in $L^q$ is integrable, with its integral controlled by the two norms. The numerical inequality behind this result is Young’s inequality.

Definition: Exponents $p,q\in(1,\infty)$ are conjugate if $1/p+1/q=1$. For such a pair, Young’s inequality states that for all $a,b\geq0$, $$ ab\leq\frac{a^p}{p}+\frac{b^q}{q}. $$

To see why Young’s inequality holds, fix $a\geq0$ and consider $\phi(b)=ab-b^q/q$ for $b\geq0$. If $a=0$, then $\phi(b)\leq0$. If $a>0$, the derivative is $\phi'(b)=a-b^{q-1}$. Thus $\phi$ increases until $b=a^{1/(q-1)}=a^{p-1}$, and decreases thereafter. Its maximum is $$ \phi(a^{p-1})=a^p-\frac{a^{(p-1)q}}{q} =a^p-\frac{a^p}{q} =\frac{a^p}{p}, $$ because $(p-1)q=p$ and $1-1/q=1/p$. Hence $ab-b^q/q\leq a^p/p$, which is Young’s inequality.

Hölder’s Inequality

Theorem (Hölder’s Inequality): Let $1<p<\infty$, and let $q$ be conjugate to $p$. If $f\in L^p(X,\mu)$ and $g\in L^q(X,\mu)$, then $fg$ is integrable and $$ \int_X |fg|\,d\mu\leq\|f\|_p\|g\|_q. $$

Proof. If either norm is zero, the zero-norm criterion from the previous tutorial shows that the corresponding function is zero almost everywhere. Therefore $fg=0$ almost everywhere, and the claimed inequality holds.

Now suppose both norms are positive. Define $F=|f|/\|f\|_p$ and $G=|g|/\|g\|_q$. These are nonnegative measurable functions satisfying $$ \int_X F^p\,d\mu=1 \qquad\text{and}\qquad \int_X G^q\,d\mu=1. $$ Young’s inequality applied pointwise gives $FG\leq F^p/p+G^q/q$. Integrating this nonnegative inequality yields $$ \int_X FG\,d\mu \leq\frac{1}{p}\int_X F^p\,d\mu +\frac{1}{q}\int_X G^q\,d\mu =\frac{1}{p}+\frac{1}{q}=1. $$ In particular, $FG$ is integrable. Since $|fg|=\|f\|_p\|g\|_q FG$, nonnegative homogeneity of the integral gives $$ \int_X|fg|\,d\mu =\|f\|_p\|g\|_q\int_X FG\,d\mu \leq\|f\|_p\|g\|_q. $$ This proves the inequality and the integrability claim.

Worked Example: Hölder’s Inequality for Two Vectors

Take $X=\{1,2\}$ with counting measure, $p=q=2$, $f(1)=1$, $f(2)=2$, $g(1)=2$, and $g(2)=-1$. Then $$ \int_X|fg|\,d\mu=|1\cdot2|+|2\cdot(-1)|=4. $$ The norms are $$ \|f\|_2=\sqrt{1^2+2^2}=\sqrt{5}, \qquad \|g\|_2=\sqrt{2^2+(-1)^2}=\sqrt{5}. $$ Thus Hölder’s estimate is $4\leq\sqrt{5}\sqrt{5}=5$. The absolute values in the product integral matter: the signed sum is $1\cdot2+2\cdot(-1)=0$, but Hölder bounds the integral of $|fg|$, not that signed sum.

The Minkowski Inequality

Theorem (Minkowski Inequality): Let $1\leq p\leq\infty$, and let $f,g\in L^p(X,\mu)$. Then $$ \|f+g\|_p\leq\|f\|_p+\|g\|_p. $$

Proof for $1<p<\infty$. By the Linear Closure of $L^p$, $h=f+g$ belongs to $L^p$. If $\|h\|_p=0$, the claimed inequality follows because its right-hand side is nonnegative. Otherwise, the pointwise triangle inequality for real numbers gives $$ |h|^p=|f+g||h|^{p-1} \leq (|f|+|g|)|h|^{p-1}. $$ Integrating and applying Hölder’s inequality to each product gives $$ \|h\|_p^p \leq\int_X|f||h|^{p-1}\,d\mu +\int_X|g||h|^{p-1}\,d\mu \leq(\|f\|_p+\|g\|_p)\,\bigl\||h|^{p-1}\bigr\|_q, $$ where $q=p/(p-1)$. The final norm is finite, and its value is $$ \bigl\||h|^{p-1}\bigr\|_q =\left(\int_X |h|^{(p-1)q}\,d\mu\right)^{1/q} =\left(\int_X |h|^p\,d\mu\right)^{1/q} =\|h\|_p^{p-1}. $$ Consequently, $\|h\|_p^p\leq(\|f\|_p+\|g\|_p)\|h\|_p^{p-1}$. Since $\|h\|_p>0$, division by $\|h\|_p^{p-1}$ proves the result.

Proof for $p=1$. The Integral Triangle Inequality gives $$ \|f+g\|_1=\int_X|f+g|\,d\mu \leq\int_X|f|\,d\mu+\int_X|g|\,d\mu =\|f\|_1+\|g\|_1. $$

Proof for $p=\infty$. The Least Almost-Everywhere Bound theorem from the previous tutorial gives $|f|\leq\|f\|_\infty$ almost everywhere and $|g|\leq\|g\|_\infty$ almost everywhere. Outside the union of the two exceptional null sets, the pointwise triangle inequality therefore gives $$ |f+g|\leq|f|+|g| \leq\|f\|_\infty+\|g\|_\infty. $$ This is an almost-everywhere bound for $|f+g|$, so the definition of the $L^\infty$ norm proves the desired inequality.

Worked Example: Comparing the Norm of a Sum with the Sum of Norms

Again use counting measure on $X=\{1,2\}$, now with $p=2$. Let $f=(1,2)$ and $g=(2,1)$, meaning $f(1)=1,f(2)=2$ and $g(1)=2,g(2)=1$. Then $$ f+g=(3,3),\qquad \|f+g\|_2=\sqrt{3^2+3^2}=\sqrt{18}. $$ Also, $\|f\|_2=\sqrt{1^2+2^2}=\sqrt{5}$ and $\|g\|_2=\sqrt{2^2+1^2}=\sqrt{5}$. Minkowski’s inequality says $$ \sqrt{18}\leq2\sqrt{5}, $$ which holds because both sides are nonnegative and $18\leq20$. In this example the inequality is strict: combining the coordinates first does not produce a norm as large as the sum of the separate norms.

Consequences and Endpoint Checks

The triangle inequality, together with definiteness and homogeneity from the previous tutorial, verifies the norm axioms for finite $p$. The same argument also shows that $f+g$ has finite $L^p$ norm; its membership was already guaranteed by linear closure, while Minkowski quantifies its size. At $p=\infty$, the estimate expresses the same principle using almost-everywhere bounds rather than integrals.

A useful consequence is the reverse triangle inequality. It says that the norm cannot change by more than the norm of the change in the function. This is often the quickest way to show that norm values vary continuously when functions are close in norm.

Corollary (Reverse Triangle Inequality): If $f,g\in L^p(X,\mu)$ for $1\leq p\leq\infty$, then $$ \bigl|\|f\|_p-\|g\|_p\bigr|\leq\|f-g\|_p. $$

Proof. Since $f=g+(f-g)$, Minkowski’s inequality gives $\|f\|_p\leq\|g\|_p+\|f-g\|_p$, and hence $\|f\|_p-\|g\|_p\leq\|f-g\|_p$. Interchanging $f$ and $g$ gives $\|g\|_p-\|f\|_p\leq\|g-f\|_p$. Homogeneity implies $\|g-f\|_p=\|f-g\|_p$. These two inequalities together give the absolute-value bound.

Worked Example: Bounding the Difference of Two Norms

On $X=\{1,2\}$ with counting measure and $p=2$, take $f=(4,1)$ and $g=(1,2)$. Their norms are $\|f\|_2=\sqrt{17}$ and $\|g\|_2=\sqrt{5}$, while $f-g=(3,-1)$ has norm $\sqrt{10}$. The reverse triangle inequality gives $$ |\sqrt{17}-\sqrt{5}|\leq\sqrt{10}. $$ Indeed, the left side is positive and $(\sqrt{17}-\sqrt{5})^2=22-2\sqrt{85}<10$, since $6<\sqrt{85}$. This verifies the estimate directly and illustrates how the distance between functions controls the difference between their sizes.

What the Inequalities Do—and Do Not—Say

Hölder’s inequality applies to a product whose factors have conjugate integrability exponents. It does not assert that every product of two $L^p$ functions is integrable with the same estimate using $p$ on both factors; the conjugate exponents are essential to the proof. Minkowski’s inequality, in turn, concerns the norm of a sum and relies on $p\geq1$. The restriction cannot simply be discarded: for exponents below $1$, the corresponding integral expression generally fails the triangle inequality.

These results are also insensitive to changes on null sets. The assumptions, products, and sums may be checked using representatives, but the resulting norms and inequalities concern equivalence classes. Whenever an argument uses pointwise inequalities, they need only hold almost everywhere for the integral or essential-supremum conclusions to follow.

Key takeaway: Hölder’s inequality controls integrals of products for conjugate exponents. Minkowski’s inequality then establishes the triangle inequality for $L^p$ norms, including the $L^1$ and $L^\infty$ endpoints, and implies the reverse triangle inequality.

Check Your Understanding

Use the definitions and inequalities developed here to answer the following questions.

What condition makes two exponents $p$ and $q$ conjugate?
In the proof of Hölder’s inequality, why are the functions divided by their norms?
Which earlier result gives the $L^1$ case of Minkowski’s inequality directly?
Why can the $L^\infty$ triangle inequality be obtained from almost-everywhere bounds?
How does the reverse triangle inequality control the change in a norm when a function is replaced by a nearby one?

What You'll Learn