## Linear Operators: General theory |

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Page 28

f : D → X be a generalized sequence of elements in a metric space X . We call f a

generalized Cauchy sequence in X , if , for each ε > 0 , there

such that eff ( p ) , f ( q ) ) < € if p 2 do , q 2 do 5 LEMMA . If f is a generalized

Cauchy ...

f : D → X be a generalized sequence of elements in a metric space X . We call f a

generalized Cauchy sequence in X , if , for each ε > 0 , there

**exists**a do e D ,such that eff ( p ) , f ( q ) ) < € if p 2 do , q 2 do 5 LEMMA . If f is a generalized

Cauchy ...

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 x a measurable function on RX R . Suppose that K

...

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

that lim f ( x )

**exists**. Putting 2 - 0 y = lub 1 ( x ) ] , OS & < 0 show that M , is a B -space . 85 Let K ( x , y ) be a à x a measurable function on RX R . Suppose that K

...

Page 362

Under the hypotheses of Exercise 37 , show that there

( x ) øn ( x ) dx if and only if the functions o n humnandin ( x ) , m 2 1 , are

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

bounded ...

Under the hypotheses of Exercise 37 , show that there

**exists**f in C with an = Sa +( x ) øn ( x ) dx if and only if the functions o n humnandin ( x ) , m 2 1 , are

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

bounded ...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero