COMPACTNESS CRITERIA AND MEASURES OF NONCOMPACTNESS IN FUNCTION SPACES

Abstract

In [9], R. S. Phillips gave a compactness criterion for subsets of a Banach space X, namely, that if (T_s) is a net of compact operators converging strongly to the identity then a bounded set A ⊆ X is relatively compact if and only if (T_s) converges uniformly on A to the identity. He then used this general result to give concrete compactness criteria in some specific spaces and to investigate compactness of operators. In this paper, we develop these ideas in two directions: firstly, to demonstrate that this method can be used as a unifying tool to derive many classical compactness criteria originally proved using other techniques, and secondly to extend the method from giving a criterion for compactness to giving a generalised measure of noncompactness. The conditions for compactness that result from this approach are not new (though they are possibly given in slightly more generality than can easily be found in the literature), but they all arise from a single, simple, and usually elementary, approach.

Throughout, vector operations applied to sets are all defined pointwise, so if A and B are subsets of a vector space, x is a vector and λ is a scalar, then

formula here

The ball centred at x with radius r in a normed space X is denoted B_X(x; r). More generally, if A is a subset of a normed space X then the r-neighbourhood B_X(A; r) of A is defined by

formula here

It is sometimes convenient to combine these notations, for example to use x+rB_X(0; 1) in place of B_X(x; r) or A+rB_X(0; 1) in place of B_X(A; r).

The ball measure of noncompactness in a normed space X will be denoted by β_X: if A is a bounded subset of X then β_X(A) is the infimum of r such that A can be covered with finitely many balls of radius r.