## Linear Operators, Part 1 |

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Page 289

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

are connected , as in Theorem 1 , by the formula * * 1 = { s } ( s ) g ( s ) u ( ds ) , fel

...

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

are connected , as in Theorem 1 , by the formula * * 1 = { s } ( s ) g ( s ) u ( ds ) , fel

...

Page 415

Hence ko e p - A , and thus pe A + k , 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 + k , 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 434

and proceed as in the first part of the

a subsequence { ym } of { { n } such that limm - * * * ym exists for each x * in the

set H of that

point ...

and proceed as in the first part of the

**proof**of the preceding theorem to constructa subsequence { ym } of { { n } such that limm - * * * ym exists for each x * in the

set H of that

**proof**. Let Rm = co { ym , Ym + 2 , . . . } and let yo be an arbitrarypoint ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 obtained 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