## Linear Operators: General theory |

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Page 28

If f is a generalized Cauchy sequence in a complete metric space X , there

a pe X such that lim f ( d ) = p . PROOF . Let dne D be such that c , c , d , implies of

( c ) , f ( c ) ) < 1 / n . Let bn be an upper bound for the finite set { dj , dq , ... , dn } ...

If f is a generalized Cauchy sequence in a complete metric space X , there

**exists**a pe X such that lim f ( d ) = p . PROOF . Let dne D be such that c , c , d , implies of

( c ) , f ( c ) ) < 1 / n . Let bn be an upper bound for the finite set { dj , dq , ... , dn } ...

Page 353

Let M be the space of all bounded 2 - measurable functions f defined on R such

that lim f ( x )

space . 85 Let K ( x , y ) be a à xâ measurable function on RX R. Suppose that K (

x ...

Let M be the space of all bounded 2 - measurable functions f defined on R such

that lim f ( x )

**exists**. Putting If = lub 11 ( x ) ] , 3-0 OS # 00 show that M , is a B -space . 85 Let K ( x , y ) be a à xâ measurable function on RX R. Suppose that K (

x ...

Page 362

Under the hypotheses of Exercise 37 , show that there

x ) & n ( x ) dx if and only if the functions - Immanen ( ) , m 2 1 , are uniformly

bounded and equicontinuous . 39 Let { an } , -00 < n < too , be a bounded

sequence ...

Under the hypotheses of Exercise 37 , show that there

**exists**f in C with an Sa * / (x ) & n ( x ) dx if and only if the functions - Immanen ( ) , m 2 1 , are uniformly

bounded and equicontinuous . 39 Let { an } , -00 < n < too , be a bounded

sequence ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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