Hölder's inequality

Hölder's inequality is the master tool for bounding the integral of a product of two functions. The intuition: if one factor is allowed to be large only where the other is small, their product cannot accumulate too much area. The inequality makes this trade-off exact by measuring the two factors in complementary norms — one in L^p, the other in L^q where p and q are conjugate exponents balancing the deal.

Precisely, let p and q satisfy 1/p + 1/q = 1 with 1 <= p, q <= infinity (the case p = 1, q = infinity included). For measurable f and g, the integral of |f g| is at most ||f||_p times ||g||_q. In particular, if f is in L^p and g is in L^q then f g is integrable. The proof rests on Young's inequality for products, ab <= a^p/p + b^q/q, applied pointwise after normalizing the norms to 1.

Equality holds precisely when |f|^p and |g|^q are proportional almost everywhere (one of them possibly zero) — that is, when the two factors are perfectly ‘aligned’. The case p = q = 2 recovers the Cauchy–Schwarz inequality. Hölder is also what shows the L^q space acts as bounded linear functionals on L^p, underpinning the duality theory.