Entropies of several sets of real valued functions

Introduction* In this paper the entropies of several sets of real valued functions are calculated. The entropy of a metric set, a notion introduced by Kolmogorov [2], is a measure of its size in terms of the minimal number of sets of diameter not exceeding 2ε necessary to cover it. The most striking use of this notion to date has been given by Kolmogorov [4] and Vituskin [7] who have shown that not all functions of n variables can be represented by functions of fewer variables if only functions satisfying certain smoothness conditions are allowed. For an exposition of this and other topics related to entropy see [5]. For other entropy calculations by the present author see [1]. The Kolmogorov-Vituskin result makes use of the following entropy calculation: