**Two converses to a refinement of the Hahn-Banach
Theorem**

by

**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”
*i _{V }*:

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* (

Counterparts of those same maps persist as well for convexoids *V*,
and our second converse is then:

* The convexoid V is* (