A Coordinate-free Foundation for Projective Spaces Treating Projective Maps from a Subset of a Vector Space into Another Vector Space
نویسندگان
چکیده
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 involutions (self-inverse permutations).
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