Condensation of the singularities in the theory of operator ideals

Abstract

In the present paper there are given some applications of the principle of condensation of the singularities of families of nonnegative functions established by W. W. Breckner in 1984. They reveal Baire category information on certain subsets of a normed linear space \(X\) of the second category that are defined by means of an inequality of the type \(f(x)<\infty\), where \(f\) is a given function from \(X\) to \([0,\infty]\). Sets of this type occur frequently in the theory of operator ideals. They are constructed individually by using entropy or approximation numbers of operators. By specializing the general results given in the paper it follows that such operator sets are of the first category, while their complements are residual \(G_\delta\)-sets, of the second category, uncountable and dense.

Keywords

• singularities of a family of real-valued functions
• residual sets
• operator ideals