## Linear Operators: General theory |

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A

A

**linear**vector space ,**linear**space , or vector space over a field Ø is an additive group X together with an operation m : 0 X X X written as m ( a , x ) = ax , which satisfy the following four conditions : ( i ) a ( x + y ) = axtay ...Page 37

X , and if T ( x + y ) = Tx + Ty , Tax ) Tx + Ty , Tlax ) = aTx for every a € ☆ and every pair x , y of vectors in the domain of T. Thus a

X , and if T ( x + y ) = Tx + Ty , Tax ) Tx + Ty , Tlax ) = aTx for every a € ☆ and every pair x , y of vectors in the domain of T. Thus a

**linear**transformation on a**linear**space X is a homomorphism on the additive group X which ...Page 410

The intersection of an arbitrary family of convex subsets of the

The intersection of an arbitrary family of convex subsets of the

**linear**space X is convex. As examples of convex subsets of X, we note the subspaces of BE, and the subsets of X consisting of one point. 3 Lemma.### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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