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Functionally closed sets and functionally convex sets in real Banach spaces | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 28، دوره 7، شماره 1، فروردین 2016، صفحه 289-294 اصل مقاله (336.81 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2015.340 | ||
| نویسندگان | ||
| Madjid Eshaghi1؛ Hamidreza Reisi Dezaki* 1؛ Alireza Moazzen2 | ||
| 1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran | ||
| 2Kosar University of Bojnord, Bojnord, Iran | ||
| تاریخ دریافت: 29 اردیبهشت 1394، تاریخ بازنگری: 15 مهر 1394، تاریخ پذیرش: 05 آذر 1394 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Convex set؛ Chebyshev set؛ Krein-Milman theorem | ||
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