{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}