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.
Eshaghi, M., Reisi Dezaki, H. & Moazzen, A. (2016). Functionally closed sets and functionally convex sets in real Banach spaces. International Journal of Nonlinear Analysis and Applications, 7(1), 289–294. https://doi.org/10.22075/ijnaa.2015.340
MLA
Eshaghi, M., Reisi Dezaki, H., & Moazzen, A. "Functionally closed sets and functionally convex sets in real Banach spaces", International Journal of Nonlinear Analysis and Applications, 7, 1, 2016, 289–294. doi: 10.22075/ijnaa.2015.340
HARVARD
Eshaghi, M., Reisi Dezaki, H., & Moazzen, A. (2016). 'Functionally closed sets and functionally convex sets in real Banach spaces', International Journal of Nonlinear Analysis and Applications, 7(1), pp. 289–294. doi: 10.22075/ijnaa.2015.340
CHICAGO
Eshaghi, M., Reisi Dezaki, H. & Moazzen, A., "Functionally closed sets and functionally convex sets in real Banach spaces." International Journal of Nonlinear Analysis and Applications, 7 1 (2016): 289–294, doi: 10.22075/ijnaa.2015.340
VANCOUVER
Eshaghi M., Reisi Dezaki H., Moazzen A. Functionally closed sets and functionally convex sets in real Banach spaces. IJNAA. 2016;7(1):289–294. doi: 10.22075/ijnaa.2015.340
International Journal of Nonlinear Analysis and Applications