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n13 - ECE 580 Math 587 Correspondence 13 SPRING 2011 Notes...

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ECE 580 / Math 587 SPRING 2011 Correspondence # 13 March 28, 2011 Notes On HAHN-BANACH THEOREM Extension Form We start with a definition – that of a sublinear functional . Definition. Let X be a real vector space (not necessarily normed). A map p : X R is called a sublinear functional if it satisfies the following two properties: p ( x + y ) p ( x ) + p ( y ) x, y X (subadditive) p ( αx ) = αp ( x ) α 0 , x X (positive homogeneity) Note that “norm” is a sublinear functional. The next result is a useful property of sublinear functionals. Lemma. Let M X be a subspace, and f a linear functional on M , such that for some sublinear functional p , f ( x ) p ( x ) x M. Let x o be a fixed element of X . Then, for any real number c , the following are equivalent : f ( x ) + λc p ( x + λx o ) x M, λ R (1) p ( x x o ) f ( x ) c p ( x + x o ) f ( x ) x M (2) Furthermore, there is a real number c satisfying (2) and hence (1) . Proof : To show that (1) (2), first set λ = 1, and then set λ = 1 and replace x by x ; then (2) is immediate. To show the converse, first take λ > 0, and replace x by x/λ on the RHS inequality, to obtain (1). Now take λ < 0, and replace x by x/λ on the LHS inequality, to arrive again at (1). For λ = 0, (1) is always satisfied, from the definitions of f and p . To produce the desired c , let x, y be arbitrary elements out of M . Then, f ( x ) f ( y ) = f ( x y ) p ( x y ) p ( x + x o ) + p ( y x o ) by subadditivity
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