## Linear Operators: General theory |

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Page 36

A linear

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

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

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

**vectors**in the domain of T. Thus ... If X is a**vector**space , if A CX , and if a is a scalar , the symbol aA is written for the set of elements of the ...Page 250

For an arbitrary

For an arbitrary

**vector**x in H the**vector**x— ( y * x ) / ( y * yılyı is in M so that ( x , y ) = y * x ( yı , y ) / y * yı = y * r , which proves the existence of the desired y . To see that y is unique , let y ' be an element of H such ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero