Why the Triangle Inequality Needs a Proof
The expression \(\|f\|_p=(\int |f|^p\,d\mu)^{1/p}\) measures the size of a function in \(L^p\), but its definition alone does not show that adding two functions preserves \(L^p\) membership or satisfies a triangle inequality. For \(1<p<\infty\), Hölder’s inequality supplies the key estimate. One subtlety matters in the proof: before establishing that \(f+g\in L^p\), we cannot simply apply Hölder to a factor involving \(|f+g|^{p-1}\) over the whole space. We will first make that factor integrable by restricting to suitable finite-measure sets, and then pass to the limit.
Let \((X,\mathcal{M},\mu)\) be any measure space; no assumption that \(\mu\) is finite, complete, or sigma-finite is needed. Functions may be real- or complex-valued and are measurable. For \(1\le p<\infty\), \(L^p(\mu)\) consists of almost-everywhere equivalence classes of functions \(f\) for which \(\int_X |f|^p\,d\mu<\infty\), with \(\|f\|_p=(\int_X|f|^p\,d\mu)^{1/p}\). We use representatives that are finite everywhere, changing them on a null set if needed. Such changes do not alter the equivalence classes or norms.
Write \(p'=p/(p-1)\), so \(1/p+1/p'=1\). We use Hölder’s inequality in the following form: if \(u\in L^p(\mu)\) and \(v\in L^{p'}(\mu)\) are measurable, then \(\int_X|uv|\,d\mu\le\|u\|_p\|v\|_{p'}\). This result applies on any measure space. The proof below uses it only after verifying that both factors belong to the required spaces.
Proof by Localization and Hölder’s Inequality
Proof. Set \(h=f+g\), which is measurable, and let \(S=\{x:|f(x)|+|g(x)|>0\}\). Outside \(S\), both \(f\) and \(g\), and hence \(h\), vanish. For each positive integer \(n\), define $$ S_n=\{x:|f(x)|+|g(x)|>1/n\},\qquad E_n=S_n\cap\{x:|h(x)|\le n\}. $$ The sets \(E_n\) increase with \(n\): both conditions defining them become weaker as \(n\) increases. Also, \(\bigcup_nE_n=S\). Indeed, a point of \(S\) has \(|f(x)|+|g(x)|>0\), so this sum exceeds \(1/n\) for all sufficiently large \(n\); and the finite value \(|h(x)|\) is at most \(n\) for all sufficiently large \(n\). No point outside \(S\) belongs to any \(E_n\).
Each \(E_n\) has finite measure. In fact, $$ S_n\subseteq\{|f|>1/(2n)\}\cup\{|g|>1/(2n)\}. $$ For any \(a>0\), the inequality \(a^p\chi_{\{|f|>a\}}\le |f|^p\) gives $$ \mu(\{|f|>a\})\le a^{-p}\int_X|f|^p\,d\mu<\infty, $$ and the same reasoning applies to \(g\). Thus \(\mu(S_n)<\infty\), and consequently \(\mu(E_n)<\infty\). Since \(|h|\le n\) on \(E_n\), the number $$ H_n=\int_{E_n}|h|^p\,d\mu $$ is finite: it is at most \(n^p\mu(E_n)\).
The scalar triangle inequality gives \(|h|\le |f|+|g|\). Multiplying by \(|h|^{p-1}\) and integrating on \(E_n\) yields $$ H_n\le\int_{E_n}|f||h|^{p-1}\,d\mu+\int_{E_n}|g||h|^{p-1}\,d\mu. $$ Hölder applies to each term. The first factors \(f\chi_{E_n}\) and \(g\chi_{E_n}\) belong to \(L^p(\mu)\), since their norms do not exceed \(\|f\|_p\) and \(\|g\|_p\). The other factor belongs to \(L^{p'}(\mu)\), because $$ \int_X\bigl(|h|^{p-1}\chi_{E_n}\bigr)^{p'}\,d\mu =\int_{E_n}|h|^p\,d\mu =H_n<\infty. $$ Its \(L^{p'}\) norm is \(H_n^{1/p'}=H_n^{(p-1)/p}\). Therefore $$ H_n\le \bigl(\|f\|_p+\|g\|_p\bigr)H_n^{(p-1)/p}. $$ If \(H_n=0\), then \(H_n^{1/p}=0\le\|f\|_p+\|g\|_p\). If \(H_n>0\), divide by \(H_n^{(p-1)/p}\) to obtain the same bound: $$ H_n^{1/p}\le\|f\|_p+\|g\|_p. $$ Now \(|h|^p\chi_{E_n}\) increases pointwise to \(|h|^p\), since \(E_n\uparrow S\) and \(h=0\) outside \(S\). By the Monotone Convergence Theorem, $$ \int_X|h|^p\,d\mu=\lim_{n\to\infty}H_n \le\bigl(\|f\|_p+\|g\|_p\bigr)^p<\infty. $$ Thus \(h\in L^p(\mu)\). Taking the \(p\)-th root of this inequality gives \(\|h\|_p\le\|f\|_p+\|g\|_p\), as required. \(\square\)
Worked Examples
Worked Example: Two Vectors on a Finite Counting Space
Take \(X=\{1,2\}\) with counting measure, \(\mathcal{M}\) the set of all subsets, and \(p=2\). Define \(f(1)=1,\ f(2)=2\) and \(g(1)=2,\ g(2)=-1\). Counting measure makes each integral the sum over the two points. Compute both sides of Minkowski’s inequality.
First, $$ \|f\|_2=\left(|1|^2+|2|^2\right)^{1/2}=(1+4)^{1/2}=\sqrt5, \qquad \|g\|_2=\left(|2|^2+|-1|^2\right)^{1/2}=(4+1)^{1/2}=\sqrt5. $$ At each point, \((f+g)(1)=1+2=3\) and \((f+g)(2)=2+(-1)=1\). Hence $$ \|f+g\|_2=(|3|^2+|1|^2)^{1/2}=(9+1)^{1/2}=\sqrt{10}. $$ The claimed bound is \(\sqrt{10}\le 2\sqrt5=\sqrt{20}\), which holds because both sides are nonnegative and \(10\le20\). Check: squaring the computed norms gives \(10\) on the left and \(20\) on the right, confirming the inequality directly.
Worked Example: A Strict Inequality for p = 3
On \(X=\{1,2\}\) with counting measure and \(p=3\), take \(f=(1,1)\) and \(g=(1,-1)\), where each ordered pair lists the values at \(1\) and \(2\). Then $$ \|f\|_3=(|1|^3+|1|^3)^{1/3}=2^{1/3}, \qquad \|g\|_3=(|1|^3+|-1|^3)^{1/3}=2^{1/3}. $$ Adding coordinate by coordinate gives \(f+g=(2,0)\), so $$ \|f+g\|_3=(|2|^3+|0|^3)^{1/3}=8^{1/3}=2. $$ Thus the inequality reads \(2\le 2\cdot2^{1/3}\). To check it exactly, both sides are positive, so cubing is order-preserving: the cubes are \(8\) and \((2\cdot2^{1/3})^3=8\cdot2=16\), respectively. Hence \(8\le16\), and the inequality is strict in this example.
Worked Example: Functions on the Unit Interval
Let \(X=[0,1]\), \(\mathcal{M}\) be the Lebesgue sigma-algebra, \(\mu=m\), and \(p=2\). Put \(f(x)=x\) and \(g(x)=1-x\). Both functions are measurable, and direct integration gives $$ \|f\|_2=\left(\int_0^1x^2\,dx\right)^{1/2} =\left(\left[\frac{x^3}{3}\right]_0^1\right)^{1/2} =\frac1{\sqrt3}. $$ For the second function, $$ \|g\|_2^2=\int_0^1(1-x)^2\,dx =\int_0^1(1-2x+x^2)\,dx =1-1+\frac13=\frac13, $$ so \(\|g\|_2=1/\sqrt3\). Their sum is the constant function \(f+g=1\), and therefore $$ \|f+g\|_2=\left(\int_0^1 1^2\,dx\right)^{1/2}=1. $$ The bound is \(1\le2/\sqrt3\). Both sides are nonnegative, and squaring reduces this to \(1\le4/3\), which is true. Check: the integral of the squared sum is \(\int_0^1 1\,dx=1\), matching the computed left-hand norm squared.
Common Pitfalls
Applying Hölder before checking its hypotheses. A tempting first line is to integrate \(|f+g|^p\le(|f|+|g|)|f+g|^{p-1}\) over all of \(X\) and immediately apply Hölder. But the second factor \(|f+g|^{p-1}\) has \(L^{p'}\) norm \((\int|f+g|^p\,d\mu)^{1/p'}\), which is precisely the quantity not yet known to be finite. That direct use of Hölder is circular. The sets \(E_n\) in the proof have finite measure and bounded \(|f+g|\), so the second factor is in \(L^{p'}\) there. The Monotone Convergence Theorem then passes from these valid local estimates to the whole space.
Using convexity in the wrong direction. Convexity of \(t\mapsto t^p\) gives $$ \left(\frac{a+b}{2}\right)^p\le\frac{a^p+b^p}{2}, $$ not the reverse inequality. For example, with \(p=2,\ a=1,\ b=\sqrt3\), the left side is $$ \left(\frac{1+\sqrt3}{2}\right)^2 =\frac{4+2\sqrt3}{4} =1+\frac{\sqrt3}{2}<2 =\frac{1+3}{2}. $$ So this convexity statement cannot be used to claim the opposite bound. When a scalar estimate of the form \((a+b)^p\ge a^p+b^p\) is needed for \(a,b\ge0\) and \(p>1\), it has a different justification: for fixed \(a\ge0\), the function \(F(b)=(a+b)^p-b^p\) has derivative \(F'(b)=p((a+b)^{p-1}-b^{p-1})\ge0\) for \(b\ge0\), and \(F(0)=a^p\). Hence \(F(b)\ge a^p\). This scalar fact is not a substitute for the localization and Hölder argument that proves Minkowski’s inequality.
Minkowski’s inequality is what makes the \(L^p\) size compatible with addition: it proves that the sum of two \(L^p\) functions remains in \(L^p\) and controls its norm. Its proof also illustrates a general discipline in integration arguments: verify the hypotheses of an inequality for the functions actually being used, rather than assuming the desired conclusion in order to apply it.
Check Your Understanding
For each question, state the measure space and exponent being used, and distinguish an equality of representatives from an equality that is only almost everywhere.
- In the proof, why does \(\mu(E_n)<\infty\) follow from \(f,g\in L^p(\mu)\)? Give the threshold-set estimate used.
- Explain why \(|h|^{p-1}\chi_{E_n}\in L^{p'}(\mu)\), and compute its \(L^{p'}\) norm in terms of \(H_n\).
- On a two-point counting space with \(p=2\), let \(u=(1,0)\) and \(v=(0,1)\). Compute \(\|u\|_2,\|v\|_2,\|u+v\|_2\), and verify Minkowski’s inequality.
- Where would a proof that applies Hölder directly to \(|f+g|^{p-1}\) on all of \(X\) risk circular reasoning?
- For \(p=2,\ a=1,\ b=\sqrt3\), calculate both sides of the midpoint convexity inequality and state which side is larger.