## Linear Operators: General theory |

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( a ) In a topological group G , any algebraic combination of any number of

variables x1 , Xn is continuous as a map of GX ... XG into G. ( b ) In a linear

and vectors ...

( a ) In a topological group G , any algebraic combination of any number of

variables x1 , Xn is continuous as a map of GX ... XG into G. ( b ) In a linear

**topological space**X , all linear combinations of any number of scalars Qi , . , c.m.and vectors ...

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non - zero linear functional on the complex space X , which separates the sets M

and N. Q.E.D. 2. Linear

1 are applied to linear

non - zero linear functional on the complex space X , which separates the sets M

and N. Q.E.D. 2. Linear

**Topological Spaces**In this section , the results of Section1 are applied to linear

**topological spaces**. Statements 1-6 of this section are ...Page 419

If I ' is a total linear space of linear functionals on X , X is a locally convex linear

maps , etc.

If I ' is a total linear space of linear functionals on X , X is a locally convex linear

**topological space**in its r topology . Note that X may already be a linear**topological space**, possessing notions of open and closed subsets , continuousmaps , etc.

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

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