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hahnban - Chapter 12 The Hahn-Banach Theorem In this...

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Chapter 12 The Hahn-Banach Theorem In this chapter V is a real or complex vector space. The scalars will be taken to be real until the very last result, the comlex-version of the Hahn-Banach theorem. 12.1 The geometric setting If A is a subset of V then the translate of A by a vector x V is the set x + A = { x + a : a A } If A and B are subsets of V and t any real number we use the notation tA = { ta : a A } and A + B = { a + b : a A, b B } If x, y V then the segment xy is the set of all points on the line running from x to y : xy = { tx + (1 - t ) y : 0 t 1 } A subset C of V is convex if for any two point P, Q C the segment PQ is contained in C . Equivalently, C is convex if λC + (1 - λ ) C C for every λ [0 , 1]. It is clear that the translate of any convex set is convex, and indeed if C is a convex set then so is a + tC for any a V and t R . 1
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2 CHAPTER 12. THE HAHN-BANACH THEOREM A subspace W of V has codimension 1 if there is a vector x V \ W such that W + R x = V . This is equivalent to saying that the quotient space V/W has dimension 1. A hyperplane is a set of the form W + x where W is any codimension one subspace and x is any vector. Let W be a codimension 1 subspace of V , and v any vector outside W . Then V can be expressed as the union of W with two open half-spaces: V = W ( W + { tv : t > 0 } ) ( W - { tv : t > 0 } ) If x is any vector in V then the hyperplane W + x specifies two closed half- spaces : W + x + { tv : t 0 } and W + x - { tv : t 0 } whose intersection is the hyperplane W + x and whose union is all of V . We shall refer to these closed half-spaces as the two sides of the hyperplane. We will prove the following geometrically intuitive fact: If C is a convex subset of V and p V a point outside C then there is a hyperplane H such that C is a subset of one side of H and p lies on the other side. Though it is possible to prove this by “purely geometric” reasoning, it will be both more convenient and more useful for our purposes to use an algebraic approach. It will be convenient to use the infinities and -∞ . We require that -∞ < , and -∞ < x < for all real numbers x . The following arithmetic operations with will be defined: t + = + t = , k = k = , 0 = 0 = 0 for all k > 0 and all t R ∪ {∞} . 12.2 The algebraic formulation Let C be a non-empty convex subset of V . If C is non-empty then we can translate C appropriately to ensure that 0 C . For this section, we assume that the origin 0 belongs to C , i.e. 0 C . The “size” of a vector v V
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12.2. THE ALGEBRAIC FORMULATION 3 relative to C is the “smallest” non-negative number t 0 such that v lies in the tC ; more precisely, define p C ( v ) = inf { t 0 : v tC } where the infimum of the empty set is taken to be . The function p C : V [0 , ] is the Minkowski functional for the set C .
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