On Constraint Qualification for an Infinite System of Convex Inequalities in a Banach Space

For a general infinite system of convex inequalities in a Banach space, we study the basic constraint qualification and its relationship with other fundamental concepts, including various versions of conditions of Slater type, the Mangasarian–Fromovitz constraint qualification, as well as the Pshenichnyi–Levin–Valadier property introduced by Li, Nahak, and Singer. Applications are given in the restricted range approximation problem, constrained optimization problems, as well as in the approximation problem with constraints by conditionally positive semidefinite functions.