## Linear Operators: General theory |

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Page 56

Since the map y → ny is a

set V . Thus , TG 2 TM - TM 2 TM - TM 2 V - V . Since a map of the form y → a - y is

a

Since the map y → ny is a

**homeomorphism**in Y , TM contains a non - void openset V . Thus , TG 2 TM - TM 2 TM - TM 2 V - V . Since a map of the form y → a - y is

a

**homeomorphism**, the set a - V is open . Since the set V - V = U ( a – V ) is the ...Page 91

However , Kolmogoroff [ 1 ] proved that a topological linear space is

convex neighborhood of the origin . Wehausen [ 1 ] showed that if a topological

linear space has ...

However , Kolmogoroff [ 1 ] proved that a topological linear space is

**homeomorphic**to a normed linear space if and only if there exists a boundedconvex neighborhood of the origin . Wehausen [ 1 ] showed that if a topological

linear space has ...

Page 442

Then there exists a

R ) with la ( r ) ] = 1 , r e R , such that [ * ] ( Tr ) ( ) = a ( r ) ( c ( r ) ) ; ? C ( Q ) , r 6 R .

Proof . The linear map T * : C * ( R ) → C * ( Q ) , defined by ( T * y * ) x = y * ( Tx )

...

Then there exists a

**homeomorphism**t between R and Q , and a function a in C (R ) with la ( r ) ] = 1 , r e R , such that [ * ] ( Tr ) ( ) = a ( r ) ( c ( r ) ) ; ? C ( Q ) , r 6 R .

Proof . The linear map T * : C * ( R ) → C * ( Q ) , defined by ( T * y * ) x = y * ( Tx )

...

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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