## Linear Operators: General theory |

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Page 57

A continuous

continuous

...

A continuous

**linear**one - to - one**map**of one F - space onto all of another has acontinuous

**linear**inverse . Proof . Let X , Y be F - spaces and T a continuous**linear**one - toone**map**with TX Y. Since ( T - 1 ) -1 = T**maps**open sets onto open...

Page 58

Nelson Dunford, Jacob T. Schwartz, William G. Bade. manifold in this product

space ; hence it is a complete metric space ( 1.6.7 ) . Thus G is an F - space . The

) .

Nelson Dunford, Jacob T. Schwartz, William G. Bade. manifold in this product

space ; hence it is a complete metric space ( 1.6.7 ) . Thus G is an F - space . The

**map**prx : [ w , Tx ] → x of G onto X is one - to - one ,**linear**, and continuous ( I.8.3) .

Page 490

Nelson Dunford, Jacob T. Schwartz, William G. Bade. is in a space of continuous

functions . In some other cases very little is known . For example , while it is easy

to see that the general continuous

Nelson Dunford, Jacob T. Schwartz, William G. Bade. is in a space of continuous

functions . In some other cases very little is known . For example , while it is easy

to see that the general continuous

**linear map**from L , [ 0 , 1 ] , p > 1 , to L [ 0 , 1 ] ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### 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