Penot - Calculus Without Derivatives

A knowledge of basic set theory is desirable for the reading of the present book, as in various branches of analysis. We assume that the reader has such a familiarity with the standard uses of set theory. But we recall here some elements related to orders, because Zorns lemma yields (among many other results) the HahnBanach theorem, which has itself numerous versions adapted to different situations.

Recall that a preorder or partial order or preference relation on a set X is a relation A between elements of X , often denoted by , with that is reflexive ( or for all ) and transitive ( i.e., for ). One also writes instead of or and one reads, y is above x or y is preferred to x . A preorder is an order whenever it is antisymmetric in the sense that for every . Two elements x , y of a preordered set are said to be comparable if either or . If such is the case for all pairs of elements of X , one says that is totally ordered . That is not always the case (think of the set of subsets of a set S with the inclusion or of a modern family with the order provided by authority). Given a subset S of , an element m of X is called an upper bound (resp. a lower bound ) of S if one has (resp. ) for all s S . A preordered set ( I ,) is directed if every finite subset F of I has an upper bound. A subset J of a preordered set ( I ,) is said to be cofinal if for all i I there exists j J such that . A map between two preordered spaces is said to be filtering if for all i I there exists such that whenever