## Linear Operators: General theory |

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Page 57

A continuous

continuous

one-to- one

A continuous

**linear**one-to-one**map**of one F-space onto all of another has acontinuous

**linear**inverse. Proof. Let X, ?) be F-spaces and T a continuous**linear**one-to- one

**map**with TX = 7). Since (T-1)-1 = T**maps**open sets onto open sets ...Page 58

The

. Hence, by Theorem 2, its inverse pf£ is continuous. Thus T = pr^pr^1 is

continuous (1.4.17). Q.E.D. 5 Theorem. // a

each of two ...

The

**map**pr^ : [x, Tx]-+ x of % onto 3E is one-to-one,**linear**, and continuous (1.8.3). Hence, by Theorem 2, its inverse pf£ is continuous. Thus T = pr^pr^1 is

continuous (1.4.17). Q.E.D. 5 Theorem. // a

**linear**space is an F-space undereach of two ...

Page 490

For example, while it is easy to see that the general continuous

„[0, 1], p > 1, to L„[0, 1] has the form no satisfactory expression for the norm of T is

known. No conditions on K (s, t) are known which are equivalent to the ...

For example, while it is easy to see that the general continuous

**linear map**from L„[0, 1], p > 1, to L„[0, 1] has the form no satisfactory expression for the norm of T is

known. No conditions on K (s, t) are known which are equivalent to the ...

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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