Compactness Criteria and Measures of Noncompactness in Function Spaces

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 A + B ={ a + b : a ∈ A , b ∈ B }, λ A ={λ a : a ∈ A }, x + A ={ x + a : a ∈ A }. 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 B x ( A ; r )={ x ∈ X : for some a ∈ A ,‖ x − a ‖< r }. 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 .