## Linear Operators, Part 1 |

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14

* with 1x * ) = 1 and x * x = 12 . PROOF . Apply Lemma 12 with Y = 0 . The æ *

required in the present

14

**COROLLARY**. For every x + 0 in a normed linear space X , there is an x * e X* with 1x * ) = 1 and x * x = 12 . PROOF . Apply Lemma 12 with Y = 0 . The æ *

required in the present

**corollary**may then be defined as x | times the * whose ...Page 188

Q . E . D . As in the case of finite measure spaces we shall call the measure

space ( S , & , u ) constructed in

, Eis ui ) the product measure space and write ( S , E , u ) = P . – 1 ( Si , Eis Mil .

Q . E . D . As in the case of finite measure spaces we shall call the measure

space ( S , & , u ) constructed in

**Corollary**6 from the o - finite measure spaces ( Si, Eis ui ) the product measure space and write ( S , E , u ) = P . – 1 ( Si , Eis Mil .

Page 662

Nelson Dunford, Jacob T. Schwartz. 2

lim , A ( n ) exists it is a projection of X upon the manifold { x \ Tx = x } of fixed

points of T . The complementary projection has the closure of ( I – T ) x for its

range .

Nelson Dunford, Jacob T. Schwartz. 2

**COROLLARY**. When the strong limit E =lim , A ( n ) exists it is a projection of X upon the manifold { x \ Tx = x } of fixed

points of T . The complementary projection has the closure of ( I – T ) x for its

range .

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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