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 $tin b(x,r)$; and $k(subseteq x)$ is functionally closed (briefly, $f$-closed), if $t(k)subseteq bbb r $ is closed for all bounded linear transformations $tin b(x,r)$. we improve the krein-milman theorem ...
