## A Hahn-Banach Theorem Converse

by
### F.E.J. Linton

Wesleyan University

Middletown, CT

06459 USA

For presentation at A.M.S. Western Section Meeting #1019,
Salt Lake City, Utah,
University of Utah, October 7-8, 2006.
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})** ) .

Here is a PDF of the
[shorter] accepted version of this abstract. Back to the
Web Home of F.E.J. Linton.

First posted: 01 August 2006. Last updated: 06 August 2006.
Copyright © 2006 F.E.J. Linton. All rights reserved.