3. Weak topologies

The existence of a minimum for an optimization problem such as min_A f is a sensitive issue in an infinite dimensional space X, since the compactness of A may be difficult to secure. The compactness fails, notably, when A is the unit ball, and it is quite possible that a continuous linear functional may not attain a minimum over B _X . What if we were to change the topology on X, in an attempt to mitigate this lack of compactness? We would want to have fewer open sets: this is called weakening the topology. The reason behind this is simple: the fewer are the open sets in a topology, the more likely it becomes that a given set is compact. On the other hand, the fewer open sets there are, the harder it is for a function defined on the space to be continuous (or lower semicontinuous), which is the other main factor in guaranteeing existence. The tension between these two contradictory pressures, the problem of finding the “right” topology that establishes a useful balance between them, is one of the great themes of functional analysis, and the subject of this chapter.