## Linear Operators: General theory |

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Page 136

There is a uniquely

There is a uniquely

**determined**smallest field and a uniquely**determined**smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family t ...Page 454

... 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

... 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

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

Let G = U_1 Yn ; then f is

Let G = U_1 Yn ; then f is

**determined**by G. It follows that if F is a denumerable subset of X * , there exists a denumerable subset G of X * such that each f € F is**determined**by Gg . We can even assert that each continuous linear ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero