(b) Let A;B[member of] WCC (X) with A [not equal to] B: We may assume without loss of generality that there exists some a [member of] A such that a [member of] B: Since B is a [T.sub.w]-compact subset of the locally convex topological vector space (X;[T.sub.w]) ; it follows from the Hahn-Banach

Separation Theorem that there exists some x* [member of] X* such that sup Rex* (b) < Rex* (a): Let [delta] = Rex* (a) [mathematical expression not reproducible] (b) > 0: We have Rex* (a) Rex* (b) [greater than or equal to] [delta] for all b [member of] B which in turn implies that

In fact, if z [not member of] C, then there exists [y.sup.*] [member of] [E.sup.*] by the

separation theorem [23] such that <z, [y.sup.*]> < [inf.sub.y [member of] C] <y,[y.sup.*]>.

Hence by the

separation theorem of convex sets, there exists ([phi], [OMEGA], [xi]) [member of] ([R.sup.m] x [S.sup.p] x [R.sup.q])\ {0, 0,0}, such that

On the other hand, as far as [mathematical expression not reproducible] in Theorem 1 is concerned, the conclusion is also directly derived from Guo et al.'s

separation theorem between a point and a [T.sub.c]-closed [L.sup.0]-convex subset in [12].

Then by the Hahn-Banach

Separation Theorem, there is [x.sup.*] [member of] [X.sup.*], [gamma] [member of] R, and [epsilon] > 0, such that

For this we need a "

separation theorem" for a finite set of points inside a [C.sup.1]-convex body.

By the classical

separation theorem, there exists a nonzero functional [z.sup.*] [member of] [Z.sup.*], such that [z.sup.*] x [greater than or equal to] [z.sup.*] y for all x [member of] K and y [member of] W.

By the Sturm type

separation theorem, one solution of (6.1) is (non)oscillatory iff every solution of (6.1) is (non)oscillatory.

The

separation theorem says, for instance, that there is a vector orthogonal to the point b' on the arm such that for any h in H, ph' [less than or equal to] ph, and for any b in B, pb' [greater than or equal to] pb.

This paper extends the Fisher

Separation Theorem of finance and microeconomic theory to include the Keynesian model of macroeconomics.

The Fisher

separation theorem of finance theory is an application of pure microeconomics and has appeared in standard general finance textbooks for years [Brealy, Myers, Sick, and Whaley, 1986].

These papers derive two major theorems: One is the "

separation theorem" which states that, when futures markets exist, the firm's export production decision is determined solely by technology and input-output prices, including the futures prices.