## Linear Operators: General theory |

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Page 36

A linear

group X together with an operation m : 0 X X →X , written as m ( a , x ) = ax ,

which satisfy the following four conditions : ( i ) a ( x + y ) = ax + ay , a EØ , x , y e

X ; ( ii ) ...

A linear

**vector space**, linear space , or**vector space**over a field Ø is an additivegroup X together with an operation m : 0 X X →X , written as m ( a , x ) = ax ,

which satisfy the following four conditions : ( i ) a ( x + y ) = ax + ay , a EØ , x , y e

X ; ( ii ) ...

Page 37

If T : X + Y and U : Y → Z are linear transformations , and X , Y , Z are linear

spaces over the same field Ø , the product ... If X is a

a is a scalar , the symbol aA is written for the set of elements of the form ax with x

in A ...

If T : X + Y and U : Y → Z are linear transformations , and X , Y , Z are linear

spaces over the same field Ø , the product ... If X is a

**vector space**, if A CX , and ifa is a scalar , the symbol aA is written for the set of elements of the form ax with x

in A ...

Page 394

Ordered spaces . There is a vast literature dealing with

also assumed to possess an additional structure of order . For example , a

partially ordered

relation x 2 ...

Ordered spaces . There is a vast literature dealing with

**vector spaces**which arealso assumed to possess an additional structure of order . For example , a

partially ordered

**vector space**is a**vector space**V in which there is defined arelation x 2 ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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 analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero