Characterizing continuous functions on compact spaces

We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.     

Banach-Saks operators on spaces of continuous functions

Summary Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm. We study in this paper Banach-Saks operators defined on C(K, E) spaces. We characterize these operators for a large class of compacts K (the scattered ones), or for a large class of Banach spaces E (the superreflexive ones). We also show by some examples that our theorems can not be extended directly.Partially supported by C.A.I.C.Y.T. grant 0338-84. The author wishes to thank Professor F.Bombal for his encouragement.     

Extensions by spaces of continuous functions

We present two complementary results on the splitting of exact sequences having the form . The first one characterizes the Banach spaces such that for every compact space . The second is a nonlinear generalization of Zippin's criterion for the extension of -valued operators.

     

Primariness of spaces of continuous functions on ordinals

Necessary and sufficient conditions on an ordinal α are given, such thatC(α) is primary, and that the general linear group ofC(α) is contractible. In particularC(α) possesses both of these properties if α is countable. This is a part of the first author's Ph. D. dissertation, prepared at The Ohio State University under the supervision of Professor W. B. Johnson. The first author was supported by a University Fellowship of The Ohio State University. This author's research was partially supported by NSF Grant MPS-74-2449.     

Distance to spaces of continuous functions

We link here distances between iterated limits, oscillations, and distances to spaces of continuous functions. For a compact space K, a uniformly bounded set H of the space of real-valued continuous functions C(K), and ε?0, we say that H ε-interchanges limits with K, if the inequality

     

Approximation of continuous functions on complex Banach spaces

Mathematical Notes - We prove a complex analog of Kurzweil’s theorem on the approximation of continuous functions on separable Banach spaces admitting a separating polynomial and obtain a...     

Dense-lineability in spaces of continuous functions

In this paper we provide a general method to prove that certain nonlinear families of continuous functions contain dense linear manifolds. An application is furnished.

     

Weighted spaces of vector-valued continuous functions

Summary We study completeness of weighted spaces of vector-valued functions. Also, the dual of a weighted space of vector-valued continuous functions on a locally compact space is characterized as a space of vector valued Radon measures with values in the topological dual of the range space. Research supported by a grant from the National Science Foundation, GP 22713. Entrata in Redazione il 25 ottobre 1970.     

Riesz spaces of real continuous functions

An Archimedean Riesz space E is isomorphic to C(X) for some completely regular Hausdorff space X if and only if there exists a weak order unit e > 0 for which E is e-uniformly complete, e-semisimple, e-separating and 2-universally e-complete.     

Multilinear isometries on spaces of vector-valued continuous functions

In this paper we study multilinear isometries defined on certain subspaces of vector-valued continuous functions. We provide conditions under which such maps can be properly represented. Our results contain all known results concerning linear and bilinear isometries defined between spaces of continuous functions. The key result is a vector-valued version of the additive Bishop’s Lemma, which we think has interest in itself.     

Orthogonally additive polynomials on spaces of continuous functions

We show that, for every orthogonally additive homogeneous polynomial P on a space of continuous functions C(K) with values in a Banach space Y, there exists a linear operator S:C(K)→Y such that P(f)=S(fn). This is the C(K) version of a related result of Sundaresam for polynomials on Lp spaces.     

Topological entropy of continuous functions on topological spaces

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems.     

A remark on spaces of bounded continuous functions

We show that C_p^*(ℚ) and C_p^*(T) are not linearly homeomorphic, thus answering a question of van Mill.