Abstract. We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π 1 1 comprehension, is needed to prove such basic facts as the existence of the weak- * closure of any norm-closed subspace of 1 = c * 0 . This is in contrast to earlier work [6,4, 7,23,22] in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π 0 2 sentences. En route to our main results, we prove the Krein-Šmulian theorem in ACA 0 , and we give a new, elementary proof of a result of McGehee on weak- * sequential closure ordinals.

Abstract. We continue the work of [14,3,1, 19,16, 4,12,11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL 0 over RCA 0 . We show that the separation theorem for separably closed convex sets is equivalent to ACA 0 over RCA 0 . Our strategy for proving these geometrical Hahn-Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.

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