## Linear Operators: General theory |

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Page 67

A closed linear manifold in a

closed linear manifold in the

equation ( ** ) y = x * y , y e Y. Clearly $ ( x * ) = 2 * , so that $ : X * → Y * . Let n : y

...

A closed linear manifold in a

**reflexive**B - space is**reflexive**. Proof . Let y be aclosed linear manifold in the

**reflexive**space X. Let 5 : x * → y * be defined by theequation ( ** ) y = x * y , y e Y. Clearly $ ( x * ) = 2 * , so that $ : X * → Y * . Let n : y

...

Page 88

conjugate is isometric was proved by Hahn [ 3 ; p . 219 ] who was ... He used the

term regular to describe what we , following Lorch [ 2 ] , have called

**Reflexivity**. The fact that the natural isomorphism of a B - space X into its secondconjugate is isometric was proved by Hahn [ 3 ; p . 219 ] who was ... He used the

term regular to describe what we , following Lorch [ 2 ] , have called

**reflexive**.Page 425

A B - space is

topology . PROOF . Let X be a

embedding of X onto X ** . Then x and x - 1 are isometries , and x maps the

closed unit ...

A B - space is

**reflexive**if and only if its closed unit sphere is compact in the weaktopology . PROOF . Let X be a

**reflexive**B - space , and let z be the naturalembedding of X onto X ** . Then x and x - 1 are isometries , and x maps the

closed unit ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero