## 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 454

... to - one mapping of K is a homeomorphism , ( cf . Lemma I . 5 . 8 ) , changing to

the X * topology does not affect the hypotheses of the lemma . We will say that a

set of continuous linear functionals F = { f } is

... to - one mapping of K is a homeomorphism , ( cf . Lemma I . 5 . 8 ) , changing to

the X * topology does not affect the hypotheses of the lemma . We will say that a

set of continuous linear functionals F = { f } is

**determined**by another set G = { g } ...Page 455

Let G = U _ Yn ; then f is

subset of X * , there exists a denumerable subset G , of X * such that each je F is

can ...

Let G = U _ Yn ; then f is

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

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero