## Linear Operators: General theory |

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Page 36

The unit of this group in a field will be written as 1 instead of e. A linear

with an operation m : $ X S -> S, written as m(a. ,r) = our, which satisfy the

following ...

The unit of this group in a field will be written as 1 instead of e. A linear

**vector****space**, linear space, or**vector space**over a field 0 is an additive group X togetherwith an operation m : $ X S -> S, written as m(a. ,r) = our, which satisfy the

following ...

Page 37

and if T(x+y) = Tx+Ty, T(ouc) = olTx for every a e 0 and every pair x, y of

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

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

and if T(x+y) = Tx+Ty, T(ouc) = olTx for every a e 0 and every pair x, y of

**vectors**inthe 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 ...

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**SB 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 | |

44 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set 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 Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero