Other Representation Theorems in Linear Spaces
In this chapter, we shall prove three representation theorems in linear spaces. The first one, the Krein-Milman theorem says that anon-void convex compact subset K of a locally convex linear topological space is equal to the closure of the convex hull of
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XII. Other Representation Theorems in Linear Spaces
for sufficiently large value of x. Hence, by (33), lim If(x}-CI~2AK,
"'~
and since A is any positive number, we get lim f(x} = C. Thus we have proved (31).
xn.
"'~
Other Representation Theorems in Linear Spaces
In this chapter, we shall prove three representation theorems in linear spaces. The first one, the Krein-Milman theorem says that anon-void convex compact subset K of a locally convex linear topological space is equal to the closure of the convex hull of the extremal points of K. The other two theorems concern the representations of a vector lattice as point functions and as set functions.
1. Extremal Points. The Krein-Milman Theorem Definition. Let K be a subset of a real- or complex-linear space X. A non-void subset M ~ K is said to be an extremal subset of K, if a proper convex combination OI.k l + (l-lX) k2 , 0 < 01. < 1, of two points kl and k 2 of K lies in M only if both kl and k 2 are in M. An extremal set of K consisting of just one point is called an extremal point of K. Example. In a three dimensional Euclidean space, the surface of a solid sphere is an extremal subset of the sphere, and every point of the surface is an extremal point of the surface . Theorem (KREIN-MILMAN) . A non-void compact convex subset K of a locally convex linear topological space X has at least one extremal point.
m
Proof. The set K is itself an extremal set of K . Let be the totality of compact extremal subsets M of K. Order IDe by inclusion relation. It is easy to see that, if IDeI is a linearly ordered subfamily of m, the nonvoid set n M is a compact extremal subset of K which is a lower bound ME\D!.
for the subfamily iml . Thus, by Zorn's lemma, IDe contains a minimal element Mo. Suppose that M o contains two distinct points Xo and Yo' Then there exists a continuous linear functional f on X such that f(x o} =1= f(yo) ' We may assume that Re 1(xo) =1= Re f(yo}' Mo being compact, the set M 1 = {x E M o; Re 1(x) = inf Re 1(y)} is a proper subset of Mo. On the other "EM.
hand, if kl and k2 are points of K such that OI.k l K. Yosida, Functional Analysis © Springer-Verlag Berlin Heidelberg 1965
+ (l-lX) k2 E M 1 for
1. Extremal Points. The Krein-Milman Theorem
363
some IX with 0 < IX < 1, then, by the extremal property of M o' hI and k2 both E Mo. It follows from the definition of M 1 that k 1 and k 2 both E MI' Hence M 1 is a closed extremal subset properly contained in Mo. Since M o is a minimal element of 9.n, we have arrived at a contradiction. Therefore M o must consist of only one point which is thus an extremal point of K . Corollary. Let K be a non-void compact convex subset of a locally convex real linear topological space X. Let E be the totality of the extremal points of K . Then K coincides with the smallest closed set containing every convex combination lX.e.(ex, ~ 0, fIX. = 1) of
f
points e.EE, i.e. , K is equal to the closure of the convex hull Conv (E) of E . Proof. The inclusion E ~ K and the convexity of K imply that Conv(E)" is containe
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