## Linear Operators: General theory |

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Page 204

Since u ( F . ) = Ssą , In ( Sam ) " , ( d8x2 ) does not converge to

5 In ( a ) Si it follows from the dominated convergence theorem ( 6.16 ) that there

is a point in So , for which n ( s ... ) is defined for all n and for which the sequence

...

Since u ( F . ) = Ssą , In ( Sam ) " , ( d8x2 ) does not converge to

**zero**and since 05 In ( a ) Si it follows from the dominated convergence theorem ( 6.16 ) that there

is a point in So , for which n ( s ... ) is defined for all n and for which the sequence

...

Page 452

If A is a subset of X , and р is in A , then there exists a non -

linear functional tangent to A at p if and only if the cone B with vertex p generated

by A is not dense in X. PROOF . If a ¢ K , then , by 2.12 we can find a functional f

and ...

If A is a subset of X , and р is in A , then there exists a non -

**zero**continuouslinear functional tangent to A at p if and only if the cone B with vertex p generated

by A is not dense in X. PROOF . If a ¢ K , then , by 2.12 we can find a functional f

and ...

Page 595

( T ) } converge to

weakly sequentially compact for each x in X , and that f vanishes at a finite set of

points of o ( T ) . If each root 20 has finite order al20 ) , if the sequences { f4m ) ...

( T ) } converge to

**zero**in the weak operator topology . Suppose that { fn ( T ) x } isweakly sequentially compact for each x in X , and that f vanishes at a finite set of

points of o ( T ) . If each root 20 has finite order al20 ) , if the sequences { f4m ) ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero