## Linear Operators: General theory |

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Page 36

is commutative if the identity ab = ba is valid . A field is a commutative ring in

which the non - zero elements form a group under multiplication . The unit of this

group in a field will be written as 1 instead of e . A

space ...

is commutative if the identity ab = ba is valid . A field is a commutative ring in

which the non - zero elements form a group under multiplication . The unit of this

group in a field will be written as 1 instead of e . A

**linear**vector space ,**linear**space ...

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and if T ( x + y ) = Tx + Ty , Tlax ) = a Tx for every a € 0 and every pair x , y of

vectors in the domain of T . Thus a

homomorphism on the additive group X which commutes with the operations of ...

and if T ( x + y ) = Tx + Ty , Tlax ) = a Tx for every a € 0 and every pair x , y of

vectors in the domain of T . Thus a

**linear**transformation on a**linear**space X is ahomomorphism on the additive group X which commutes with the operations of ...

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2 LEMMA . ( 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

dy , .

2 LEMMA . ( 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 scalarsdy , .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 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 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