Some Extensions of the Tietze-Urysohn Theorem

Abstract

In this note we examine certain sharpenings of the Tietze-Urysohn extension theorem. We prove, for example, that if A is a closed subspace of a normal space X then there is a continuous extender v,:C*(A) -C*(X), where C*(A) and C*(X) are the Banach spaces of continuous, bounded, real-valued functions on A and X, respectively. However, well-known examples show that we cannot arrange that q is both continuous and linear, or that q is an isometry. Introduction. The classical Tietze-Urysohn theorem guarantees that every continuous, bounded, real-valued function f defined on a closed subspace A of a normal space X can be extended to a continuous, real-valued function f defined on all of X and having the same bounds as f. If we let C*(A) and C*(X) denote the real vector spaces of continuous, bounded, real-valued functions on A and X respectively, then the Axiom of Choice yields the following slight reformulation of the classical theorem. A. THEOREM (Tietze-Urysohn). If A is a closed subspace of the normal space X then there is a function r: C*(A ) C*(X) such that for every f E C*(A), rl(f) extends f and has the same bounds as f Any function q:C*(A)-+ C*(X) having the property that 71(f) extends f whenever f E C*(A) will be called an extender from C*(A) to C*(X). It is natural to wonder whether, if one selects the extension 71(f) of f more carefully, one can obtain an extender r1 which respects the natural linear and/or topological structures of the function spaces in Theorem A. For metric spaces, an affirmative answer is provided by the Dugundji Extension Theorem. B. THEOREM [D]. Suppose A is a closed subset of a metrizable space X. Then there is a function 77: C*(A) -* C*(X) such that: (1) for each f E C*(A), r(f) extends f and has the same bounds as f; (2) the function 77 is linear.(2' Theorem B is a model for all of the extension theorems considered in this paper so we shall pause to make a few comments about its conclusions and its proof. In introducing the theorem we said that it shows that 7q can be constructed in such a way that 7q respects the natural topological structures of the function spaces. Let us now introduce one such structure, the sup-norm topology. For a function f E C*(A) the sup-norm of f is defined by lIf || = sup {if(x){:x E A}; the norm of a function F E C*(X) is analogously defined. When equipped with these norms, C*(A) and C*(X) are Banach spaces and the two conclusions of Theorem B force 77 to be an isometry. That may be seen as follows. Let f, g E C*(A). Then 1j 71(f) 71(g) jj = 1177 (f g) I since 77 is linear, and because 7q(f g) has the same bounds as f g, 1j 71 (f g) jj = jj f g 1j. Thus 7q is a linear, isometric extender. The proof of Theorem B is too technical to include here but its central idea is easily described, at least in principle. One does not extend the functions in C*(A) one at a time as in the proof of Theorem A; instead one defines an extension process which acts simultaneously (and linearly) on all members of C*(A). Dugundji's extension process has been improved in recent years so that it now 1. This paper was completed while the third author was visiting the University of Pittsburgh as a Mellon Postdoctoral Fellow. 2. Dugundji actually proved that a linear q can be found such that for each f E C*(A), v(f) extends f and the range of q (f) is contained in the convex hull of the range of f. Furthermore, if one considers bounded, continuous, complex-valued functions, then one can obtain complex-linear extenders with the property that the range of v1(f) is contained in the convex hull of the range of f.