## Linear Operators: General theory |

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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 /•'-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 /•'-space. Further, a

normed linear space ...

Page 347

(b) Show that l^iS, E, fi) is

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

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

(b) Show that l^iS, E, fi) is

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

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

Page 663

Then T maps (i-measurable functions into fx-measurable functions and fi-

linear map of the F-space M(S) = M(S, E, fi, X) of all X- valued /i-measurable

functions into ...

Then T maps (i-measurable functions into fx-measurable functions and fi-

**equivalent**functions into ^-**equivalent**functions. Furtliermore T is a continuotislinear map of the F-space M(S) = M(S, E, fi, X) of all X- valued /i-measurable

functions into ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

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### Common terms and phrases

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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set 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 Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero