For 1 < p < ∞ with 1/p + 1/q = 1, Hölder's inequality states ∫|fg| ≤ ‖f‖_p ‖g‖_q. This is the key inequality enabling duality in L^p theory. When p = q = 2, it reduces to Cauchy–Schwarz.
You already know the Lᵖ norm ‖f‖_p = (∫|f|ᵖ)^(1/p), which measures the "size" of a function in a way that weights extreme values more heavily as p grows. Hölder's inequality answers the question: how large can the integral of the product |fg| be, given that you know ‖f‖_p and ‖g‖_q? The answer is the elegant bound ∫|fg| ≤ ‖f‖_p ‖g‖_q — and the condition 1/p + 1/q = 1 is precisely what makes this tight.
The pair (p, q) satisfying 1/p + 1/q = 1 are called conjugate exponents. When p = 2, we get q = 2 as well, and Hölder's inequality becomes the Cauchy–Schwarz inequality ∫|fg| ≤ ‖f‖₂ ‖g‖₂ — a fact you may recognize from inner product spaces. Hölder generalizes this: for p = 3 and q = 3/2, Hölder's inequality handles situations where f has L³ integrability but g is only in L^(3/2). The conjugate condition is not arbitrary; it emerges from the proof via Young's inequality (ab ≤ aᵖ/p + bq/q for a, b ≥ 0), which in turn follows from the convexity of the exponential function.
The proof strategy is instructive: normalize by replacing f with f/‖f‖_p and g with g/‖g‖_q, reducing to the case where both norms are 1 and you need to show ∫|fg| ≤ 1. Then apply Young's inequality pointwise to get |f(x)g(x)| ≤ |f(x)|ᵖ/p + |g(x)|^q/q, and integrate both sides. The right-hand side integrates to 1/p + 1/q = 1. This argument reveals why the conjugate condition is necessary: it is exactly what makes Young's inequality integrate to 1.
The deepest significance of Hölder's inequality is its role in Lᵖ duality. Every bounded linear functional on Lᵖ can be represented as integration against some function in Lq — written symbolically as (Lᵖ)* ≅ Lq. This duality underpins the Riesz representation theorem and is central to functional analysis. Hölder's inequality is what makes this identification *bounded*: without it, you couldn't control ∫fg by separate norms. It also immediately implies the Minkowski inequality (triangle inequality for Lᵖ norms), which is the next step in the theory.