Twisted Fréchet Spaces of Continuous Functions

Abstract

We will give an example of a Fnkhet space of continuous functions (with the compact· open topology) which is not isomorphic to a product of Banach spaces. The author has met fairly general theorems showing that some spaces of continuous functions with the compact·open topology (esp. Frechet spaces of that type) arc isomor· phic to topological products of Banach spaces [10], but, on the other hand , he has never found any published counterexample, i.e., any example of a Fr&:het space of continuous functions which is not isomorphic to a product of Banach spaces. H seems impossible that such an example has not been known but . .. such a space would be also an example of a twisted (i.e., non·isomorphic to a product of spaces with continuous norms) Frechet space. The first twisted Frechet space was constructed only in 1980 by V.B. Moscatelli [9] and the introduced method turned out to be a fruitful source of counterexamples [9] and [1] . In fact, we prove that applying Moscatelli's method in a suitable manner we can obtain a space of conti nuous functions. The author is very indebted to Prof. V. 8 . Moscatelli for the remarks concering the first version of the paper. Let us recall that a topological space X is O·dimensional iff every point in X has a. neighbourhood base consisting of closed·open sets. We call X a k·space if for every topological space Y a function f : X ......Y is continuous whenever all its restrictions to compact subsets are continuous (for more details see [4, eh. 3.3]). We always consider spaces of continuous fuctions C(X) equipped with the compact·open topology. This space is complete whenever X is a k·space [6,3.6.4 ). It is easily seen that C(X) is metrizable (normable) iff there is a fundamental sequence of compact sets in X (X is compact) and, in that case, C(X) is separable iff there is a weaker Hausdorff mctrizable topology 011 X [6,2. 1O.3J.