## Linear Operators: General theory |

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Results 1-3 of 73

Page 2

The set A is said to be a proper subset of the set B if AC B and A + B . The

notation ACB , or B A , will

of a set A relative to a set B is the set whose elements are in B but not in A , i . e . ,

the ...

The set A is said to be a proper subset of the set B if AC B and A + B . The

notation ACB , or B A , will

**mean**that A is a proper subset of B . The complementof a set A relative to a set B is the set whose elements are in B but not in A , i . e . ,

the ...

Page 261

The space rba ( S ) is partially ordered by defining u 2 a to

E ) for every E in the field determined by the closed sets . Similarly , the space C (

S ) is partially ordered by defining 1 Zg to

The space rba ( S ) is partially ordered by defining u 2 a to

**mean**that u ( E ) 2 2 (E ) for every E in the field determined by the closed sets . Similarly , the space C (

S ) is partially ordered by defining 1 Zg to

**mean**that f ( s ) 2 g ( s ) for all s in S ...Page 660

These results are applied to obtain , in Section 7 , corresponding

pointwise convergence theorems for the continuous case . Section 8 is

concerned with a certain class of operators T for which the sequence of averages

{ N - 1 EN ...

These results are applied to obtain , in Section 7 , corresponding

**mean**andpointwise convergence theorems for the continuous case . Section 8 is

concerned with a certain class of operators T for which the sequence of averages

{ N - 1 EN ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero