We use of two notions functionally Convex (briefly, F-Convex) and functionally closed (briefly, F-closed) in functional analysis and obtain more results. We show that if {Aa}aÎI is a family F- Convex subsets with non empty intersection of a Banach space X, then ÈaÎI Aa is F–Convex. Moreover, we introduce new definition of notion F-Convexiy.

Application of the network equivalent concept for external system representation for power system transient analysis is well known. However, the challenge to utilize an equivalent network, approximated by a rational function, is to guarantee the passivity of the corresponding model. In this regard, special techniques are required to enforce the passivity of the equivalent model through a post processing approach that minimizes its impact on the original model characteristics. In this paper, the passivity is enforced by expressing the problem in terms of a Convex optimization problem that guarantees the global optimal solution. The Convex optimization problem is efficiently solved by recently developed numerical interior–point methods. This passivity enforcement is also global which indicates that the passivity enforcement in one region does not lead to passivity violation in other regions.

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Let X be a real normed space, then C( X) is functionally Convex (brie y, F-Convex), if T(C)  R is Convex for all bounded linear transformations T 2 B(X; R); and K( X) is functionally closed (brie y, F-closed), if T(K)  R is closed for all bounded linear transformations T 2 B(X; R). We improve the Krein-Milman theorem on  nite 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 (brie y, 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.