EXTREME POINTS OF THE UNIT BALL IN THE DUAL SPACES OF

SOME REAL SUBSPACES OF C(X) by S.H.Kulkarni, Madras, and B.V.Limaye, Bombay, India

Abstract. Let A be a nonzero subspace of a normed space C over the real or the complex numbers. If ϕ belongs to the closed unit ball L(A) of the dual of A, then conditions are given for ϕ to be an extreme point of L(A) in terms of Hahn-Banach extensions of ϕ to C and the extreme points of C. These consideration are applied to the case where C = C(X)_{r}, the normed space of all complex-valued continuous functions on X regarded as a real linear space. In particular, we consider a continuous involution τ on X, let

C(X, τ) = {f ∈ C(X)_{r} : f ∘ τ = f̄}

and treat (real) subspace of C(X, τ). A characterization of the extreme points of L(C(X, τ)) (given by Grzesiak) is deduced. If A is a real subspace of C(X, τ) and ϕ ∈ L(A), then the uniqueness of the representing measures for ϕ is related to ϕ being an extreme point of L(A). Mathematics subject classification: 46A22, 46B22, 46E15, 28C05. Keywords: Extreme points, dual of a normed space, Hahn-Banach theorem, Krein-Milman theorem, continuous involution, representing measure.