## Linear Operators: General theory |

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Page 50

( 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

Olom ...

( 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 Oz , . . . ,Olom ...

Page 413

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

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

Section 1 are applied to linear

section ...

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 ofSection 1 are applied to linear

**topological spaces**. Statements 1 - 6 of thissection ...

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 1 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

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero