## 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 250

For an arbitrary

) = y ** ( y1 , y ) / y * yı y * x ( y1 , y ) / y * yı = y * x , which proves the existence of

the desired y . To see that y is unique , let y ' be an element of H such that y * x ...

For an arbitrary

**vector**æ in H the**vector**x- ( y * x ) / ( y * yı ) yı is in M so that ( x , y) = y ** ( y1 , y ) / y * yı y * x ( y1 , y ) / y * yı = y * x , which proves the existence of

the desired y . To see that y is unique , let y ' be an element of H such that y * x ...

Page 318

last section permit us to develop a more satisfactory theory of

countably additive set functions ( briefly ,

able ...

**Vector**Valued Measures The theorems on spaces of set functions proved in thelast section permit us to develop a more satisfactory theory of

**vector**valuedcountably additive set functions ( briefly ,

**vector**valued measures ) than we wereable ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

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