By Leonard D. Berkovitz

A textbook for a one-semester starting graduate path for college students of engineering, economics, operations learn, and arithmetic. scholars are anticipated to have a great grounding in easy actual research and linear algebra.

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Therefore, there exists an a " 0 such that 1a, x2 - 0 for all x in C 9 y. Hence for all x in C, 1a, x 9 y2 - 0, and so 1a, x2 - 1a, y2 for all x in C. SEPARATION THEOREMS 53 If we now let : 1a, y2, we get the ﬁrst conclusion of the theorem. 16. 2. 3. L et X and Y be two disjoint convex sets. a that separates them. Note that the theorem does not assert that proper separation can be achieved. 2 in the sequel will allow us to conclude that the separation is proper. 3 does not yield this fact. To prove the theorem, we ﬁrst note that X 5 Y : ` if and only if 0 , X 9 Y.

Let A : A 6 A . Then A is closed, but co(A) : +(x , x ) : x 9 0, is open.      We now prove the lemma. 2 L> co(A) : x : x :  p x x + A p : (p , . . , p ) + P . 1 the set is L> compact in RK, where m:(n;1);n(n;1):(n;1). The mapping : RK ; RL deﬁned by L> (p, x , . . , x ) :  p x  L G G G is continuous. 2 that compact. 2, ( ) : co(A), so co(A) is compact. 8. If O is an open subset of RL, then co(O) is also open. Proof. Since O 3 co(O), the set co(O) has nonempty interior.

X ) whose coordinates satisfy a x ; % ; a x : 0  L   L L for some nonzero vector a : (a , . . , a ). Thus, if : 0 and a " 0, then the  L hyperplane Ha passes through the origin and is an (n 9 1)-dimensional subspace of RL. In Section 10 of Chapter I we saw that for a given linear functional L on RL there exists a unique vector a such that L(x) : 1a, x2 for all x in RL. This representation of linear functionals and the deﬁnition of the hyperplane Ha? show that hyperplanes are level surfaces of linear functionals.