## Linear Operators, Part 1 |

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Page 67

A closed linear manifold in a

closed linear manifold in the

equation $ ( w * ) y = x * y , y e Y . Clearly 1 $ ( w * ) ] = $ x * l , so that £ : X * → 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 6 : x * * y * be defined by theequation $ ( w * ) y = x * y , y e Y . Clearly 1 $ ( w * ) ] = $ x * l , so that £ : X * → Y *

.

Page 72

19 If X is a

Exercise 18 , that both 3 and X / 3 are

and both 3 and X / 3 are

* ...

19 If X is a

**reflexive**B - space , and 3 is a closed subspace of X , show , usingExercise 18 , that both 3 and X / 3 are

**reflexive**. 20 If X is a B - space , with 3 CX ,and both 3 and X / 3 are

**reflexive**, then X is**reflexive**. ( Hint : Given any * * * e X ** ...

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero