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Page 424
Since each projection is a continuous map , each o the sets A ( x , y ) and B ( a , x ) is closed . Hence tK = Nx , veX A ( x , y ) onae , tex B ( a , x ) is also closed . Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the ...
Since each projection is a continuous map , each o the sets A ( x , y ) and B ( a , x ) is closed . Hence tK = Nx , veX A ( x , y ) onae , tex B ( a , x ) is also closed . Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the ...
Page 429
( Krein - Šmulian ) A convex set in X * is X - closed if and only if its intersection with every positive multiple of the closed unit sphere of X * is X - closed . Proof . This follows from the preceding theorem and Corollary 2.14 .
( Krein - Šmulian ) A convex set in X * is X - closed if and only if its intersection with every positive multiple of the closed unit sphere of X * is X - closed . Proof . This follows from the preceding theorem and Corollary 2.14 .
Page 488
It follows from the definition of U * that every element in its range satisfies the stated condition . Q.E.D. 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a closed range , then UX = Y. PROOF .
It follows from the definition of U * that every element in its range satisfies the stated condition . Q.E.D. 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a closed range , then UX = Y. PROOF .
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero