Functionally closed sets and functionally convex sets in real Banach spaces

Document Type: Research Paper


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

2 Kosar University of Bojnord, Bojnord, Iran


‎Let $X$ be a real normed  space, then  $C(\subseteq X)$  is  functionally  convex  (briefly, $F$-convex), if  $T(C)\subseteq \Bbb R $ is  convex for all bounded linear transformations $T\in B(X,R)$; and $K(\subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)\subseteq \Bbb R $ is  closed  for all bounded linear transformations $T\in B(X,R)$. 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 $X\times \mathbb{R}$. We show that every  function $f : (a,b)\longrightarrow \mathbb{R}$ which has no  vertical asymptote is $F$-convex.