## Linear Operators: General theory |

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Page 48

The first occurrence of a maximum principle

principle (as in Theorem 2.6) is in Hausdorff [1; p. 140]. Zorn [1] gave a theorem

essentially

p.

The first occurrence of a maximum principle

**equivalent**to the well-orderingprinciple (as in Theorem 2.6) is in Hausdorff [1; p. 140]. Zorn [1] gave a theorem

essentially

**equivalent**to Theorem 2.7. A similar theorem is due to R. L. Moore [1;p.

Page 91

admits an

invariant metric and is complete under each invariant metric. Thus every

complete linear metric space can be metrized to be an .F-space. Further, a

normed linear space ...

admits an

**equivalent**metric under which it is complete, then G admits aninvariant metric and is complete under each invariant metric. Thus every

complete linear metric space can be metrized to be an .F-space. Further, a

normed linear space ...

Page 347

(b) Show that L1("Sr, Z, fi) is

collection of atoms of finite measure {£„} in Z such that every measurable subset

of S— \J„=1E„ is either an atom of infinite measure or a null set. 50 Show that no ...

(b) Show that L1("Sr, Z, fi) is

**equivalent**to \ if and only if there exists a countablecollection of atoms of finite measure {£„} in Z such that every measurable subset

of S— \J„=1E„ is either an atom of infinite measure or a null set. 50 Show that no ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero