## Linear Operators, Part 1 |

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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 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 311

According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S,

such that B(S, 2) is

isometric isomorphism a' -– a between B” (S. X) and bass, X'), which is

determined by the ...

According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S,

such that B(S, 2) is

**equivalent**to C(S). Theorem 5.1 shows that there is anisometric isomorphism a' -– a between B” (S. X) and bass, X'), which is

determined by the ...

Page 347

(b) Show that L1(S, X, u) is

collection of atoms of finite measure {En} in X such that every measurable subset

of S – U. 1 E, is either an atom of infinite measure or a null set. 50 Show that no ...

(b) Show that L1(S, X, u) is

**equivalent**to l, if and only if there exists a countablecollection of atoms of finite measure {En} in X such that every measurable subset

of S – U. 1 E, is either an atom of infinite measure or a null set. 50 Show that no ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero