One of the most important theorem in analysis is the HABY-BANACH theorem ([SJ, pi). [186][187][188][189][190][191][192][193][194][195][196][197]. The analytic form of this theorem can be written as follows:
Let S be a linear space over the field of real numhers R, and let p : X -R be a wldinear (subadditive and non-negatively homogeneous) functional on X. Let &I he a linear subspace of X, and let f : &!-R he a linear functional on M such that f ; p N , i.e.. z ~M = f ( z ) s p ( x ) . Then there exists a linear functionalf: X + R on S such that: (1)fis an extension o f f , i.e., zEM*f((r)=f(x). and (2) f i s majorized
I)y p , i.e.,f>p, or x ~X * j ( z ) s p ( z ) .
A number of important geometric consequences can he drawn from the HAHN-BAXAVH theorem. These are mainly of a separation nature saying that disjoint sets of certain types can be separated by hyperplanes. For example, consider the geometric form of the HAHN-BANACH theorem-originalIy established by S.
~I A Z U R :
Let (X, 8) he a topological vector space over the real or complex field K.
Let A be a non-empty, convex, open subset of X ; and let J4 be an affine subspace of S such that ATIM=0. Then there exists a closed affine hyperplane H in X such that X & H and H n A = 0 . In this theorem, as in most applications of the theory of convexity (see [6]) one deals with open (or clo~ed) Convex Nets and with closed hyperplanes.
J n this paper, we attempt t o relax these topological restrictions hy using simple iiotioiifi of sequential convergence. First, we o1)tain a geometric form of the HAHN-BANACR theorem which deals with sequentially open convex sets and with sequent ially closed affine hyperplanes. Then a number of other "sequential" separation theorems are obtained. Since locally convex topological vector spaces-which have the property that every open neighborhood of the zero vector 0 contains a convex open neighhorhood of 0-are the most useful and important topological vector spaces; one might expect, that sequentially 10e~all?j convez topological oector spaces (S-locally convex topobgiml vector spares)-defined M I as to have the property that every sequen-1) Some of the results are contained in the author's Ph. D. thesis written a t the University of Virginia under t,he direction of Professor E. J. &fCSH.PNE.