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A COORDINATE-FREE FOUNDATION FOR PROJECTIVE SPACES TREATING PROJECTIVE MAPS
 

Summary: A COORDINATE-FREE 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 cross-ratio.
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 one-dimensional
projective spaces of such type may be described quite simply in a geometric manner using invo-
lutions (self-inverse 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

  

Source: Ariyawansa, Ari - Department of Mathematics, Washington State University

 

Collections: Mathematics