This report outlines how a mild refinement of the Hahn-Banach Theorem
serves to characterize the real Banach spaces from amongst the abstract algebras
whose operations are the natural operations, finitary and infinitary, on
Banach discs. These algebras, which constitute the varietal reflection
of the (non-varietal) category of Banach spaces, are what Semadeni, Pumplün,
and others have called (complete) convexoids; the operations are all possible
“sub-convex-combination” operators arising from absolutely summable real sequences (finite or infinite) with (l (1) ) norm
The Hahn-Banach Theorem, commonly
read as asserting that the natural
Fortunately, counterparts of these three maps remain available even for V a mere convexoid, and our converse is then:
The convexoid V is (the unit disc of) a real Banach space if (and, of course, only if) the map iV is an equalizer of the pair
( iV** , (iV)** ) .
Here is a PDF of the [shorter] accepted version of this abstract. Back to the Web Home of F.E.J. Linton.