Minkowski's Inequality

For the $\ell^p$ norm to deserve the name norm, it must satisfy the triangle inequality: the length of a sum is at most the sum of the lengths. That statement is Minkowski’s inequality, and it is the property that makes $\ell^p$ a genuine normed space. The proof is a clever splitting followed by a single application of Hölder’s inequality.

The theorem

For $p \geq 1$, write the $\ell^p$ norm of a finite sequence as $\|a\|_p \coloneqq \big(\sum_i |a_i|^p\big)^{1/p}$.

Theorem (Minkowski’s Inequality). Let $p \geq 1$ and let $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ be real (or complex) numbers. Then

$$\left(\sum_{i=1}^{n} |a_i + b_i|^{p}\right)^{1/p} \;\leq\; \left(\sum_{i=1}^{n} |a_i|^{p}\right)^{1/p} + \left(\sum_{i=1}^{n} |b_i|^{p}\right)^{1/p}, \tag{1}$$

that is, $\|a + b\|_p \leq \|a\|_p + \|b\|_p$.

The two easy exponents

Case $p = 1$. Inequality $(1)$ is just the ordinary triangle inequality summed termwise: $|a_i + b_i| \leq |a_i| + |b_i|$, so $\sum_i |a_i + b_i| \leq \sum_i |a_i| + \sum_i |b_i|$.

So assume $p > 1$ for the rest, and let $q = \dfrac{p}{p-1}$ be its conjugate exponent, so that $(p-1)q = p$ and $\dfrac{1}{p} + \dfrac{1}{q} = 1$.

Proof for $p > 1$: split, then apply Hölder

Let $S \coloneqq \sum_{i=1}^{n} |a_i + b_i|^{p}$. If $S = 0$ both sides of $(1)$ are zero, so assume $S > 0$.

Split the power. Write one factor of $|a_i + b_i|$ off the front and bound it with the triangle inequality $|a_i + b_i| \leq |a_i| + |b_i|$:

$$|a_i + b_i|^{p} = |a_i + b_i|\,\cdot\,|a_i + b_i|^{p-1} \;\leq\; |a_i|\,|a_i + b_i|^{p-1} + |b_i|\,|a_i + b_i|^{p-1}.$$

Summing over $i$,

$$S \;\leq\; \underbrace{\sum_{i=1}^{n} |a_i|\,|a_i + b_i|^{p-1}}_{(\mathrm{I})} \;+\; \underbrace{\sum_{i=1}^{n} |b_i|\,|a_i + b_i|^{p-1}}_{(\mathrm{II})}. \tag{2}$$

Apply Hölder to each piece. Treat $(\mathrm{I})$ as the pairing of the sequence $(|a_i|)$ with $(|a_i + b_i|^{p-1})$ and apply Hölder’s inequality with exponents $p$ and $q$:

$$(\mathrm{I}) \;\leq\; \left(\sum_{i=1}^{n} |a_i|^{p}\right)^{1/p} \left(\sum_{i=1}^{n} |a_i + b_i|^{(p-1)q}\right)^{1/q}.$$

Because $(p-1)q = p$, the second factor is $\big(\sum_i |a_i + b_i|^{p}\big)^{1/q} = S^{1/q}$. Hence

$$(\mathrm{I}) \;\leq\; \|a\|_p \, S^{1/q}, \qquad\text{and likewise}\qquad (\mathrm{II}) \;\leq\; \|b\|_p \, S^{1/q}.$$

Combine. Substituting both bounds into $(2)$,

$$S \;\leq\; \big(\|a\|_p + \|b\|_p\big)\, S^{1/q}.$$

Since $S > 0$, divide both sides by $S^{1/q}$. Using $1 - \dfrac{1}{q} = \dfrac{1}{p}$,

$$S^{1/p} = S^{\,1 - 1/q} \;\leq\; \|a\|_p + \|b\|_p,$$

which is exactly $(1)$. $\square$

Why this matters

Minkowski’s inequality is the triangle inequality for the $\ell^p$ norm. Together with the easy facts $\|a\|_p \geq 0$ (with equality only for the zero sequence) and $\|\lambda a\|_p = |\lambda|\,\|a\|_p$, it certifies that $\|\cdot\|_p$ is a genuine norm — and therefore that $\ell^p$ is a normed vector space for every $p \geq 1$. The same argument, with sums replaced by integrals, gives Minkowski’s inequality for $L^p$ function spaces.

Summary

• Minkowski’s inequality: $\|a + b\|_p \leq \|a\|_p + \|b\|_p$ for $p \geq 1$ — the triangle inequality for the $\ell^p$ norm.
• Case $p = 1$ is the termwise triangle inequality; the work is in $p > 1$.
• Proof for $p > 1$: split $|a_i + b_i|^p = |a_i + b_i|\cdot|a_i + b_i|^{p-1}$, bound the first factor by $|a_i| + |b_i|$, and apply Hölder’s inequality to each resulting sum; the exponent identity $(p-1)q = p$ makes the leftover factor collapse to $S^{1/q}$.
• Consequence: $\|\cdot\|_p$ is a norm, so $\ell^p$ (and $L^p$) are normed spaces.