## 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 ...Page 58

The

. 8 . 3 ) . Hence , by Theorem 2 , its inverse prą is continuous . Thus T = pry prą ?

is continuous ( I . 4 . 17 ) . Q . E . D . 5 THEOREM . If a

The

**map**prx : [ X , Tx ] → x of ( onto X is one - to - one ,**linear**, and continuous ( 1. 8 . 3 ) . Hence , by Theorem 2 , its inverse prą is continuous . Thus T = pry prą ?

is continuous ( I . 4 . 17 ) . Q . E . D . 5 THEOREM . If a

**linear**space is an F ...Page 490

Nelson Dunford, Jacob T. Schwartz. 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

di g ...

Nelson Dunford, Jacob T. Schwartz. 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 ] has the formdi g ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero