SEPARABLE BANACH-SPACE THEORY NEEDS STRONG SET EXISTENCE AXIOMS

We investigate the strength of set existence axioms needed for separab le Banach space theory. We show that a very strong axiom, Pi(1)(1) com prehension, is needed to prove such basic facts as the existence of th e weak- closure of any norm-closed subspace of l(1) = c(0)*. This is in contrast to earlier work [6, 4, 7, 23, 22] in which theorems of sep arable Banach space theory were proved in very weak subsystems of seco nd order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Pi(2)(0) sentences. En route to our main resu lts, we prove the Krein-Smulian theorem in ACA(0), and we give a new, elementary proof of a result of McGehee on weak- sequential closure o rdinals.