For a compact metric space X , consider a linear subspace A of C{X) containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if A separates points, then the closure of A under both minima and maxima is dense in C{X). By the Hahn-Banach Theorem, if A separates probability measures, A is dense in C{X). It is shown that if A separates points from probability mea...

This lemma was then used the following year as a crucial step in the proof of the well-known Bishop-Phelps theorem [1] that every Banach space is subreflexive; in other words, every functional on a Banach space E can be approximated by a norm-attaining functional on the same space. The original proof of this lemma uses the Hahn-Banach theorem and is therefore fairly abstract. In this note, we p...

We show that for a given C*-algebra A and for any pair of Hilbert A-modules {{M, 〈., .〉},N ⊆ M} every bounded A-linear mapping r : N → A can be continued to a bounded A-linear mapping r : M → A so that (i) ‖r‖ = ‖r‖, (ii) r restricted to N equals r and (iii) the extended mappings of N ′ form a Banach A-submodule of M wherein the extensions {r n : n ∈ N} of the standardly embedded mappings {rn =...

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact we establish a slightly stronger version which applies to a wider class of boundary 1-cycles. Arguments here are based on the Hahn-Banach Theorem and some geometric m...

We investigate the concepts of linear convexity and C-convexity in complex Banach spaces. The main result is that any C-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a C-convex domain Ω in the Banach space X and a point p / ∈Ω, there is a complex hyperplane through p that does not intersect Ω. We also prov...

We investigate the concepts of linear convexity and C-convexity in complex Banach spaces. The main result is that any C-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a C-convex domain Ω in the Banach space X and a point p / ∈Ω, there is a complex hyperplane through p that does not intersect Ω. We also prov...

In this paper we proved a nonconvex separation property for general sets which coincides with the Hahn-Banach separation theorem when sets are convexes. Properties derived from the main result are used to compute the subgradient set to the distance function in special cases and they are also applied to extending the Second Welfare Theorem in economics and proving the existence of singular multi...

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C n which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact we establish a slightly stronger version which applies to a wider class of boundary 1-cycles. Arguments here are based on the Hahn-Banach Theorem and some geometric...

In this paper, we generalize Fuzzy Banach contraction theorem establishedby V. Gregori and A. Sapena [Fuzzy Sets and Systems 125 (2002) 245-252]using notion of altering distance which was initiated by Khan et al. [Bull. Austral.Math. Soc., 30(1984), 1-9] in metric spaces.