## Linear Operators: General theory |

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Page 36

A linear

group X together with an operation m : 0 X 3£ -+ 3£, written as ;/i(oc, x) = cnx,

which satisfy the following four conditions: (i) x(x+y) = auc+xy, <xe#, x,ye£; (ii) (x+

fl)x ...

A linear

**vector space**, linear space, or**vector space**over a field 0 is an additivegroup X together with an operation m : 0 X 3£ -+ 3£, written as ;/i(oc, x) = cnx,

which satisfy the following four conditions: (i) x(x+y) = auc+xy, <xe#, x,ye£; (ii) (x+

fl)x ...

Page 37

and if T(x-Sry) = Tx+Ty, T{auv) — olTx for every ae$ and every pair x, y of

in the domain of T. Thus a linear transformation on a linear

homomorphism on the additive group 36 which commutes with the operations of

...

and if T(x-Sry) = Tx+Ty, T{auv) — olTx for every ae$ and every pair x, y of

**vectors**in the domain of T. Thus a linear transformation on a linear

**space**36 is ahomomorphism on the additive group 36 which commutes with the operations of

...

Page 394

Ordered spaces. There is a vast literature dealing with

also assumed to possess an additional structure of order. For example, a partially

ordered

Ordered spaces. There is a vast literature dealing with

**vector spaces**which arealso assumed to possess an additional structure of order. For example, a partially

ordered

**vector space**is a**vector space**93 in which there is defined a relation x ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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### Common terms and phrases

a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero