 
Summary: A COORDINATEFREE FOUNDATION FOR
PROJECTIVE SPACES TREATING PROJECTIVE MAPS
FROM A SUBSET OF A VECTOR SPACE INTO
ANOTHER VECTOR SPACE
K. A. ARIYAWANSA, W. C. DAVIDON AND K. D. McKENNON
Abstract
In a related paper [2], the authors have shown that a homeomorphism which preserves
convex sets, mapping an open subset of one locally convex topological vector space onto an open
subset of another, is a projective map (the quotient of an affine operator by an affine functional).
The establishment of this result in its full generality required a treatment of (possibly infinite
dimensional) topological projective spaces. The present paper supplies this treatment, developing
projective spaces as dual pairs by virtue of the crossratio.
The nontopological portion of the paper is valid for projective spaces over (commutative)
fields of characteristic different from 2. It is demonstrated in particular, that onedimensional
projective spaces of such type may be described quite simply in a geometric manner using invo
lutions (selfinverse permutations).
1. Introduction
Let V and X be Hausdorff, locally convex, real, topological vector spaces
of dimension greater than 1. Motivated by an application [1] in the derivation
of optimization algorithms, Ariyawansa, Davidon and McKennon [2] show that a
