## 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 nietrizcd 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 nietrizcd to be an F-space. Further, a

normed linear space ...

Page 347

(b) Show that £^(5, E, fi) is

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

of S— L)^iE„ is either an atom of infinite measure or a null set. 50 Show that no ...

(b) Show that £^(5, E, fi) is

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

of S— L)^iE„ is either an atom of infinite measure or a null set. 50 Show that no ...

Page 663

Then T maps fi-measurable functions into /n-measurable functions and fi-

linear map of the F-space M(S) = M(S, E, ft, X) of all devalued fi-measurable

functions into ...

Then T maps fi-measurable functions into /n-measurable functions and fi-

**equivalent**functions into fi-**equivalent**functions. Furthermore T is a continuouslinear map of the F-space M(S) = M(S, E, ft, X) of all devalued fi-measurable

functions into ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero