Rings of functions with certain Lipschitz properties

Charles Harris Scanlon

Vol. 32 (1970), No. 1, 197–201

DOI: 10.2140/pjm.1970.32.197

Abstract

Let (X, d) denote a metric space, L_e(X) the ring of real valued functions on X which are Lipschitz on each compact subset of X,L₁(X) the ring of real valued functions on X which are locally Lipschitz relative to the completion of X, and L_c^∗(X),L₁^∗(X) the bounded elements of L_c(X),L₁(X). The relations between equality of these rings and the topological properties of X are studied. It is shown that a subspace (S,d) of (X, d) is L_c-embedded (or L_c^∗-embedded) in (X, d) if and only if S is closed. Further, every subspace of (X, d) is L_1⁻ and L₁^∗-embedded in (X,d).

Mathematical Subject Classification

Primary: 46.55

Milestones

Received: 3 April 1969

Published: 1 January 1970