Lipschitz spaces

Abstract

If (S, d) is a metric space and 0 < α < 1, Lip (S, dα) is the Banach space of real or complex-valued functions f on S such that ∥ f ∥ =max(∥ f ∥∞, ∥ f ∥dα) < ∞, where ∥ f ∥dα = sup (│ f(s) − f(t) │d-α (s, t): s≠t). The closed subspace of functions f such that limd(s, t)→0 │f(s) — f(t)│ d-α (s, t) = 0 is denoted by lip (S, dα). The main result is that, when infs≠t d(s, t) = 0 lip (S, dα) con­tains a complemented subspace isomorphic with c0 and Lip (S, d) contains a subspace isomorphic with l∞. From the construc­tion, it follows that lip (S, dα) is not isomorphic to a dual space nor is it complemented in Lip (S, dα). © 1974 Pacific Journal of Mathematics.