Let be a real normed space, then is functionally convex (briefly, -convex), if is convex for all bounded linear transformations ; and is functionally closed (briefly, -closed), if is closed for all bounded linear transformations . 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 on is functionally convex (briefly, -convex) if epi is a -convex subset of . We show that every function which has no vertical asymptote is -convex.
Eshaghi, M. , Reisi Dezaki, H. and 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. doi: 10.22075/ijnaa.2015.340
MLA
Eshaghi, M. , , Reisi Dezaki, H. , and 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
M. Eshaghi , H. Reisi Dezaki and A. Moazzen, "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. International Journal of Nonlinear Analysis and Applications, 2016; 7(1): 289-294. doi: 10.22075/ijnaa.2015.340