## Linear Operators: General theory |

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Page 58

Hence , by Theorem 2 , its inverse prą ' is

each of two metrics , and if one of the corresponding topologies contains the

other ...

Hence , by Theorem 2 , its inverse prą ' is

**continuous**. Thus T = pry prz ' is**continuous**( I.4.17 ) . Q.E.D. 5 THEOREM . If a**linear**space is an F - space undereach of two metrics , and if one of the corresponding topologies contains the

other ...

Page 417

If a

which has an interior point , then the functional is

A ...

If a

**linear**functional on a**linear**topological space separates two sets , one ofwhich has an interior point , then the functional is

**continuous**. Proof . Let X be a**linear**topological space , and A1 , A , C X. Let h be a**linear**functional separatingA ...

Page 418

If Ky , K , are disjoint closed , convex subsets of a locally convex linear

topological space X , and if K , is compact , then some non - zero

convex subset of ...

If Ky , K , are disjoint closed , convex subsets of a locally convex linear

topological space X , and if K , is compact , then some non - zero

**continuous****linear**functional on X separates K , and Kg . 12 COROLLARY . If K is a closedconvex subset of ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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 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