## Linear Operators: General theory |

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Page 22

in A has a neighborhood containing at most

number of these neighborhoods cover A, the sequence {an} consists of only

in A has a neighborhood containing at most

**a finite**number of an. Since**a finite**number of these neighborhoods cover A, the sequence {an} consists of only

**a****finite**number of distinct points of A, and therefore most certainly does have a ...Page 245

Every linear operator on

Proof. Let {bv . . ., bn) be a Hamel basis for the

space X so that every a; in $ has a unique representation in the form x = ol^-^.

Every linear operator on

**a finite**dimensional normed linear space is continuous.Proof. Let {bv . . ., bn) be a Hamel basis for the

**finite**dimensional normed linearspace X so that every a; in $ has a unique representation in the form x = ol^-^.

Page 556

Spectral Theory in

denote

Our aim is to study the algebraic and topological properties of T. This is achieved,

...

Spectral Theory in

**a Finite**Dimensional Space Throughout this section, £ willdenote

**a finite**dimensional complex JB-space, and T a linear operator in JB(X).Our aim is to study the algebraic and topological properties of T. This is achieved,

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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

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