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 ≤ 1 .

The Hahn-Banach Theorem, 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 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 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 i_V is an equalizer of the pair ( i_V** , (i_V)** ) .

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