## Linear Operators: General theory |

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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

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 -

( 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

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 .### What people are saying - Write a review

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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