Abstract

We have introduced two new notions of convexity and closedness in functional analysis. Let X be a real normed space, then <TEX>$C({\subseteq}X)$</TEX> is functionally convex (briefly, F-convex), if <TEX>$T(C){\subseteq}{\mathbb{R}}$</TEX> is convex for all bounded linear transformations <TEX>$T{\in}B$</TEX>(X, R); and <TEX>$K({\subseteq}X)$</TEX> is functionally closed (briefly, F-closed), if <TEX>$T(K){\subseteq}{\mathbb{R}}$</TEX> is closed for all bounded linear transformations <TEX>$T{\in}B$</TEX>(X, R). By using these new notions, the Alaoglu-Bourbaki-Eberlein-<TEX>${\check{S}}muljan$</TEX> theorem has been generalized. Moreover, we show that X is reflexive if and only if the closed unit ball of X is F-closed. James showed that for every closed convex subset C of a Banach space X, C is weakly compact if and only if every <TEX>$f{\in}X^{\ast}$</TEX> attains its supremum over C at some point of C. Now, we show that if A is an F-convex subset of a Banach space X, then A is bounded and F-closed if and only if every element of <TEX>$X^{\ast}$</TEX> attains its supremum over A at some point of A.