L-infinity

L^infinity is the home of functions that have a genuine ceiling — almost everywhere. It is the limiting case of the L^p family as p tends to infinity, where the norm stops averaging and instead reads off the largest value the function effectively attains. A function belongs here precisely when it stays below some finite bound except possibly on a set of measure zero, which the theory cheerfully ignores.

Formally, L^infinity(X, mu) consists of all measurable functions f whose essential supremum of |f| is finite, and the L^infinity norm ||f||_infinity is exactly that essential supremum. As with the other L^p spaces, elements are equivalence classes modulo equality almost everywhere. The space is a Banach space — complete under its norm — and L^infinity is the dual of L^1 when the measure is sigma-finite.

Two cautions. First, the name ‘infinity’ refers to the exponent, not the values: L^infinity functions are bounded (essentially), not large. Second, L^infinity is markedly less friendly than the finite-p spaces: it is not separable in general and, unlike L^p for finite p, it is not the dual of anything natural in the usual duality scheme — the dual of L^infinity is strictly larger than L^1. For finite-measure spaces, L^infinity sits inside every L^p.