dual of Lp

The dual of a space is the collection of all the ways to take a linear ‘reading’ of its elements — every continuous linear functional that turns a vector into a number. For L^p the striking fact is that these readings are not exotic: each one is just ‘pair against a fixed function and integrate’. So the dual of L^p turns out to be another L-space, namely L^q with q the conjugate exponent. The abstract dual is concretely identified with familiar functions.

Precisely, for 1 <= p < infinity and a sigma-finite measure, every bounded linear functional T on L^p has the form T(f) = the integral of f times g for a unique g in L^q, where 1/p + 1/q = 1; moreover the operator norm of T equals ||g||_q. This sets up an isometric isomorphism between the dual space (L^p)* and L^q. Hölder's inequality is exactly what makes each such integral a bounded functional, and the Radon–Nikodym theorem is the engine that produces the representing g.

The endpoint p = infinity is the famous exception. The dual of L^1 is L^infinity (for sigma-finite measures), but the dual of L^infinity is strictly larger than L^1 — it contains extra ‘finitely additive’ functionals that no integrable function can represent. So the clean reflexive symmetry ((L^p)*)* = L^p holds for 1 < p < infinity but fails at both endpoints 1 and infinity.