On compactness in functional analysis

Abstract

The theorem of Arzel& and Ascoli, characterizing conditionally compact subsets of the Banach space C(X) of continuous functions defined on a compact topological space X, is fundamental for much of functional analysis. Of less importance but still of interest is the question of characterizing subsets of C(X) which are conditionally compact in other naturally chosen topologies, such as the weak topology of C(X) as a Banach space, or the topology of pointwise convergence. This problem was considered in the case that X= [0, 1 ] by G. Sirvint [19; 20](2). It is the purpose of this paper to treat the general case; in doing so we adopt quite a different point of view. We shall find it convenient to make use of the notions of universal nets (here called U-nets) introduced by J. L. Kelley [13] and of quasi-uniform convergence due to C. Arzela [1]. Since previous familiarity with these concepts is not assumed, in ??1 and 2 we state the properties that will be needed. In order to have a wide range of applicability, we have chosen to present ?3 in an abstract formulation which is specialized in later parts. It is felt that this treatment brings out clearly the r6le of quasi-uniform convergence; further, it emphasizes the duality inherent in these compactness theorems, but which is not usually made explicit. This duality was suggested by R. S. Phillips [17] and a form of it was employed systematically by V. Smulian [21 ] for bounded subsets of a separable Banach space. The general formulation was made by S. Kakutani [12] for the case of uniform convergence, in much the same vein as here. The results of ?3 are applied, in ?4, to certain subsets of a Banach space in order to fix these ideas and to indicate how readily they permit symmetric proofs of the classical theorems of [Gantmacher](2) Schauder concerning [weakly] compact operators and their adjoints. In ?5, we prove that a necessary and sufficient condition that a collection