Vol. 32, No. 1, 1970

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Rings of functions with certain Lipschitz properties

Charles Harris Scanlon

Vol. 32 (1970), No. 1, 197–201

Let (X, d) denote a metric space, Le(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 Lc(X),L1(X) the bounded elements of Lc(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 Lc-embedded (or Lc-embedded) in (X, d) if and only if S is closed. Further, every subspace of (X, d) is L1 and L1-embedded in (X,d).

Mathematical Subject Classification
Primary: 46.55
Received: 3 April 1969
Published: 1 January 1970
Charles Harris Scanlon