
Organizers 
Two converses to a refinement of the HahnBanach
Theorem
by
Fred E.J. Linton
Wesleyan University, Middletown, CT,
USA [Emeritus]
A littleknown mild refinement of the HahnBanach Theorem helps to characterize the real Banach spaces both from amongst the real normed linear spaces, as well as from amongst the abstract algebras whose operations are the natural operations, finitary and infinitary, on Banach discs. These algebras, sometimes called convexoids, constitute the varietal reflection of the category of Banach spaces; the natural operations referred to are the various “subconvexcombination” operators arising from all the absolutely summable real sequences (finite or infinite) with (l^{(1)}) norm ≤ 1.
The HahnBanach Theorem itself, commonly read as asserting that the natural “evaluation map” i_{V}: V → V^{**} from any real Banach space V to its second dual V^{**} is an isometric embedding, is easily tweaked to reveal (and this is that mild refinement) that i_{V} is actually an equalizer (or “difference kernel”) of the corresponding evaluation map i_{(V**)}: V^{**} → V^{****} for V^{**} and the second transpose (i_{V})^{**}: V^{**} → V^{****} of i_{V} itself.
Fortunately, counterparts of these maps persist when V is merely a real normed linear space, and our first converse (to the tweaked HahnBanach Theorem) is then:
The real normed linear space V is complete (i.e., is already a real Banach space) if (and only if) the map i_{V} is an equalizer of the pair ( i_{(V**)} , (i_{V})^{**} ).
Counterparts of those same maps persist as well for convexoids V, and our second converse is then:
The convexoid V is (the unit disc of) a real Banach space if (and only if) the map i_{V} is an equalizer of the pair ( i_{(V**)} , (i_{V})^{**} ).
Date received: October 28, 2007