Content: Two converses to a refinement of the Hahn-Banach Theorem.

Usual basic Hahn-Banach Theorem asserts the existence of enough continuous linear functionals on Banach space V to separate points of V. Usual consequence: evaluation map i_V: V → V^** is an isometric embedding.

A	→	B
↓		↓
A^**	→	B^**

But if f : A → B is bounded linear transformation, the square shown commutes; and if f is an isometric embedding, that square is actually a pullback diagram (A = B ∩ A^** in B^**).

V	→	V^**
↓		↓
V^**	→	V^****

Because i_V is an isometric embedding, then, the next diagram is also a pullback diagram, displaying Banach space V as equalizer (or difference kernel) of i_(V^**) and (i_V)^**: V^** → V^**** .

Even for just normed linear spaces V, the conjugate spaces V^** and V^**** are still Banach spaces; and any equalizer of two continuous linear transformations defined on a Banach space remains (complete, hence) a Banach space, whence:
Converse 1: The normed linear space V is a Banach space if (and only if) i_V is the equalizer of the two maps i_(V^**) and (i_V)^**: V^** → V^**** .

Similar considerations apply when V denotes a formal complete convexoid, i.e., one of 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 unit disk of each Banach space is such an algebra, via ordinary “convergent infinite series” considerations; moreover, the formal conjugate of each such algebra turns out in fact to be (the unit disk of) a real Banach space. And the equalizer of two convexoid-homomorphisms between two (unit disks of) Banach spaces happens to remain (the unit disk of) a Banach space [NB: here not only the source object but also the target must be (the unit disk of) a Banach space]. Consequently:

Converse 2: The formal complete convexoid V is (the unit disk of) a Banach space if (and only if) i_V is the equalizer of the two maps i_(V^**) and (i_V)^**: V^** → V^**** .

Reference ~~~~~~~~~ Graves (editor): Conference on Integration, Topology, and Geometry in Linear Spaces, UNC, May 1979. In: Contemporary Mathematics, Vol. 2, Amer. Math. Soc., 1980 (cf. especially pp. 227-240).
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