FUNCTIONAL ANALYTIC AXIOMS AND SET-THEORY

In analysis the Axiom of Choice and some of its consequence are used to obtain non-measurable subsets of R, non-continuous linear functionals on Banach spaces, and to prove that on every Banach space E not equal {0} there exists non-zero linear continuous functionals. These examples cannot be obtained in a constructive way. The weaker axiom of dependent choice is sufficient for most results of analysis. We show that using the weaker axiom we may add some very simple new axioms (incompatible with the axiom of choice but consistent with the other axioms of Zermelo-Frankel) which imply the following properties: (a) Every linear functional on a Banach space is continuous; (b) Every linear operator from a Banach space into another is continuous; (c) On the Banach space l(infinity)/Co zero is the only linear continuous functional. These properties follow, respectively, from the following axioms: (A) Every linear functional on el is continuous; (B) Every norm (or seminorm) on l(1) is continuous; (C) l'(infinity) = l(1) (i.e., every linear continuous functional on l(infinity) is defined by an element of l(1)). We mention several open problems.