## Linear Operators: General theory |

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Page 136

6 LEMMA . There is a uniquely

at least one field , namely the field of all subsets of S , which contains a given

family t .

6 LEMMA . There is a uniquely

**determined**smallest field and a uniquely**determined**smallest o - field containing a given family of sets . PROOF . There isat least one field , namely the field of all subsets of S , which contains a given

family t .

Page 455

Let G = U21 Yn ; then / is

subset of X * , there exists a denumerable subset Ge of X * such that each fe F is

can ...

Let G = U21 Yn ; then / is

**determined**by G . It follows that if F is a denumerablesubset of X * , there exists a denumerable subset Ge of X * such that each fe F is

**determined**by GF We can even assert that each continuous linear functional fcan ...

Page 657

... determinism whereby it is known that an initial state x will , after t seconds have

elapsed , have passed into a uniquely

uniquely

) .

... determinism whereby it is known that an initial state x will , after t seconds have

elapsed , have passed into a uniquely

**determined**new state y . Since y isuniquely

**determined**by x and t , a function , on S to S is defined by the equa$ :( «) .

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### 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