Continuous linear extension of functions

Abstract

Abstract. Let (X, d) be a complete metric space. We prove that there is a continuous, linear, regular extension operator from the space C * b of all partial, continuous, real-valued, bounded functions with closed, bounded domains in X to the space C * (X) of all continuous, bounded, real-valued functions on X with the topology of uniform convergence on compact sets. This is a variant of a result of Kunzi and Shapiro for continuous functions with compact, variable domains.

On continuous linear operators extending metrics

“…The set of continuous, bounded functions on X is denoted by C * (X). Using the same reasoning as in Theorem 4.1 we obtain a different proof of the main result from [11]:…”

“…In particular, Kunzi and Shapiro [12] proved that there exists a continuous, regular, linear operator extending continuous, real functions defined on compact subsets of a metric space. A variant of the Kunzi-Shapiro result for the noncompact case was obtained in [11] (see also [5]). Tymchatyn and Zarichnyi obtained an analogue of the Kunzi-Shapiro theorem for (pseudo)metrics [15].…”

“…In return, the norm of the operator increases, but it can be chosen so that its norm is arbitrarily close to 1. We also obtain a different proof of the result from [11] on simultaneous, linear extensions of real functions defined on closed and bounded subsets of a complete metric space.…”