Hölder's Inequality

∑ |a_i b_i| ≤ |a|_p |b|_q for conjugate p, q — Cauchy–Schwarz generalized to L^p.

The idea

Theorem (Hölder's inequality).

Let $p, q > 1$ be conjugate exponents, meaning $\dfrac{1}{p} + \dfrac{1}{q} = 1$. Then for any real sequences $a$ and $b$, $\sum_i |a_i b_i| \;\leq\; \left(\sum_i |a_i|^{p}\right)^{1/p} \left(\sum_i |b_i|^{q}\right)^{1/q} \;=\; \|a\|_p \, \|b\|_q,$ with equality if and only if the sequences $|a_i|^{p}$ and $|b_i|^{q}$ are proportional.

The quantity $\|a\|_p$ is the $p$-norm of $a$: raise the absolute value of each entry to the power $p$, sum, and take the $p$-th root of the total. Hölder's inequality therefore bounds a sum of products by two separate measurements of size, one of $a$ alone and one of $b$ alone, taken with different exponents.

The exponents cannot be chosen independently: the conjugate condition ties them together, and increasing $p$ forces $q$ down toward $1$. A norm that weighs the large entries of $a$ more heavily is paired with one that weighs the large entries of $b$ less heavily.

Ways to work on it

• Walkthrough. Verify Hölder's inequality on a small pair of vectors and see how Cauchy–Schwarz drops out as a special case.
• Proof. See why — Young's inequality applied to unit-normalized sequences, then rescaled.
• Practice. Find the conjugate exponent q for a random p.
• Hardest. Compute the Hölder slack for random small vectors.

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