Conjugate Spaces and Universal Spaces

Abstract

Motivated by the importance of conjugate spaces indicated in earlier sections, we devote the bulk of this final chapter to some further considerations regarding such spaces. We begin with the famous Riesz-Kakutani characterization of C(Ω, ℝ)* as the space of regular signed Borel measures on Ω. After giving some applications of this theorem we proceed to some characterizations of general conjugate spaces, and use these to exhibit some new conjugate spaces (spaces of operators and Lipschitz functions). The fact that certain spaces of operators are conjugate spaces has some interesting implications for optimization theory as we shall see. We shall also establish an isomorphism between certain spaces of Lipschitz functions and certain spaces of L⁰⁰ type. A particular consequence of this is an example of a pair of Banach spaces (namely, ℓ¹(ϰ₀) and L¹([0, 1])) which fail to be isomorphic, yet whose conjugate spaces are isomorphic.