Functionally closed sets and functionally convex sets in real Banach spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

2 Kosar University of Bojnord, Bojnord, Iran

Abstract

‎Let X be a real normed  space, then  C(X)  is  functionally  convex  (briefly, F-convex), if  T(C)R is  convex for all bounded linear transformations TB(X,R); and K(X)  is  functionally   closed (briefly, F-closed), if  T(K)R is  closed  for all bounded linear transformations TB(X,R). We improve the    Krein-Milman theorem  on finite dimensional spaces. We partially prove the Chebyshev 60 years old open problem. Finally, we introduce  the notion of functionally convex  functions. The function f on X is  functionally convex (briefly, F-convex) if epi f is a F-convex subset of X×R. We show that every  function f:(a,b)R which has no  vertical asymptote is F-convex.

Keywords

Volume 7, Issue 1 - Serial Number 1
January 2016
Pages 289-294
  • Receive Date: 19 May 2015
  • Revise Date: 07 October 2015
  • Accept Date: 26 November 2015