Lipschitz spaces

Jerry Alan Johnson

Vol. 51 (1974), No. 1, 177–186

DOI: 10.2140/pjm.1974.51.177

Abstract

If (S,d) is a metric space and 0 < α < 1, Lip (S,d^α) is the Banach space of real or complex-valued functions f on S such that ∥f∥ = max(∥f∥_∞,∥f∥_a^a) < ∞, where ∥f∥_a^α = sup{|f(s) −f(t)|d^−α(s,t) : s≠t}. The closed subspace of functions f such that lim_d(s,t)→0|f(s) −f(t)|d^−α(s,t) = 0 is denoted by lip (S,d^α). The main result is that, when inf _s≠td(s,t) = 0 lip (S,d^α) 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,d^α) is not isomorphic to a dual space nor is it complemented in Lip (S,d^α).

Mathematical Subject Classification 2000

Primary: 46E15

Milestones

Received: 29 November 1972

Revised: 8 March 1973

Published: 1 March 1974