No abstract

Abstract. Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space X to L p (µ, X), for 0 < p < ∞. Local versions of these properties and results are also considered.

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B = B(X, V ) the space of all bounded linear operators from X into V; let P(X, V ) be the set of all projections in B. For a given cone S ⊂ X, we denote by P = P S (X, V ) the set of operators P ∈ P such that P S ⊂ S. When P S = ∅, we characterize those P ∈ P S for which P is minimal. This characterization is then utilized in several applications and examples.

Ž. We study the complex strongly extreme points of bounded subsets of continuously quasi-normed vector spaces X over ‫.ރ‬ When X is a complex normed linear Ž . space, these points are the complex analogues of the familiar real strongly extreme points. We show that if X is a complex Banach space then the complex strongly extreme points of B admit several equivalent formulations some of which X are in terms of ''pointwise'' versions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of B and B where 0 -p -ϱ, X and each X , j g I, are complex l Ž X .

No abstract