## Linear Operators, Part 1 |

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Page 15

( b ) For every pair of distinct points x and y , there are

and y . ( c ) For every closed set A , and every x ˘ A , there are

neighborhoods of A and x . ( d ) For every pair of

there are ...

( b ) For every pair of distinct points x and y , there are

**disjoint**neighborhoods of xand y . ( c ) For every closed set A , and every x ˘ A , there are

**disjoint**neighborhoods of A and x . ( d ) For every pair of

**disjoint**closed sets A and B ,there are ...

Page 320

The semi - variation of the vector valued measure u is defined by Ilve | | ( E ) =

sup | I Q ; 4 ( E : ) ) , EeŁ , where the supremum is taken over all finite collections

of scalars with Jai S1 and all partitions of E into a finite number of

E ...

The semi - variation of the vector valued measure u is defined by Ilve | | ( E ) =

sup | I Q ; 4 ( E : ) ) , EeŁ , where the supremum is taken over all finite collections

of scalars with Jai S1 and all partitions of E into a finite number of

**disjoint**sets inE ...

Page 461

and an arbitrary convex set is possible , provided they are

Theorem 2 . 8 ) . He also proved that a convex set K which is compact in the X *

topology of the normed linear space X , can be separated from an arbitrary

closed ...

and an arbitrary convex set is possible , provided they are

**disjoint**( compareTheorem 2 . 8 ) . He also proved that a convex set K which is compact in the X *

topology of the normed linear space X , can be separated from an arbitrary

closed ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

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