## Linear Operators: General theory |

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Page 311

1 shows that there is an isometric

ba ( S , E ) , which is determined by the equation w * XE = u ( E ) , E € E . Thus ,

since B ( S , E ) is equivalent to C ( S ) , ba ( S , E ) is equivalent to rca ( S ) ...

1 shows that there is an isometric

**isomorphism**æ * t u between B * ( S , E ) andba ( S , E ) , which is determined by the equation w * XE = u ( E ) , E € E . Thus ,

since B ( S , E ) is equivalent to C ( S ) , ba ( S , E ) is equivalent to rca ( S ) ...

Page 312

Let S , be a compact Hausdorff space such that B ( S , E ) is isometrically

( EUF ) = T ...

Let S , be a compact Hausdorff space such that B ( S , E ) is isometrically

**isomorphic**with C ( Si ) . ... The correspondence Xe → Xe , establishes an**isomorphism**t of the field onto the field of all open and closed sets in Sy , i . e . , 7( EUF ) = T ...

Page 313

Recalling that t is an

isometric

TE : ) ) = sup I IME : ) ) = \ vel i = 1 where { Ej , . . . , En } is an arbitrary partition of S

...

Recalling that t is an

**isomorphism**of onto £ , it is clear that the mapping T is anisometric

**isomorphism**of ba ( S , E ) onto ba ( S , , £j ) , since Tul = sup | ( Tu ) (TE : ) ) = sup I IME : ) ) = \ vel i = 1 where { Ej , . . . , En } is an arbitrary partition of S

...

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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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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