Two converses to a refinement of the Hahn-Banach
Fred E.J. Linton
Wesleyan University, Middletown, CT, USA [Emeritus]
A little-known mild refinement of the Hahn-Banach 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 “sub-convex-combination” operators arising from all the absolutely summable real sequences (finite or infinite) with (l (1)) norm ≤ 1.
The Hahn-Banach Theorem itself, commonly read as asserting that the natural “evaluation map” iV : 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 iV is actually an equalizer (or “difference kernel”) of the corresponding evaluation map i(V**): V** → V**** for V** and the second transpose (iV)**: V** → V**** of iV itself.
Fortunately, counterparts of these maps persist when V is merely a real normed linear space, and our first converse (to the tweaked Hahn-Banach 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 iV is an equalizer of the pair ( i(V**) , (iV)** ).
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 iV is an equalizer of the pair ( i(V**) , (iV)** ).[ >> ]