## Linear Operators: General theory |

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Page 289

so that there is a functional y * € L * such that w * * ( * ) = y * ( g ) when g and x *

are connected , as in Theorem 1 , by the formula * * 1 = [ st ( 8 ) g ( s ) u ( ds ) ...

**Proof**. Let æ * * € ( L * ) * . By Theorem 1 , LTM is isometrically isomorphic to Lq ,so that there is a functional y * € L * such that w * * ( * ) = y * ( g ) when g and x *

are connected , as in Theorem 1 , by the formula * * 1 = [ st ( 8 ) g ( s ) u ( ds ) ...

Page 415

Hence ko e p - A , and thus pe A + ko CA + K . Q . E . D . Since the commutativity

of the group G is not essential to the

Abelian topological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X

: ( i ) ...

Hence ko e p - A , and thus pe A + ko CA + K . Q . E . D . Since the commutativity

of the group G is not essential to the

**proof**, the same result holds for non -Abelian topological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X

: ( i ) ...

Page 699

The

11 as outlined in the diagram : C = CPx + DPE - Dx = Ck . The

implication CPx = DP which is the

of the ...

The

**proof**of this lemma is the most involved of all the steps in the**proof**of Lemma11 as outlined in the diagram : C = CPx + DPE - Dx = Ck . The

**proof**of theimplication CPx = DP which is the

**proof**of Lemma 14 is very similar to the**proof**of the ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero