## Linear Operators: General theory |

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Page 57

A continuous

continuous

one-to- one

Theorem 1 ) ...

A continuous

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

**linear**inverse. Proof. Let X, 3) be F-spaces and T a continuous**linear**one-to- one

**map**with TX = ?J. Since = T**maps**open sets onto open sets (Theorem 1 ) ...

Page 490

is in a space of continuous functions. In some other cases very little is known. For

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

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

is in a space of continuous functions. In some other cases very little is known. For

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

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

Page 664

dition (ii) shows that T maps //-equivalent functions into //-equivalent functions, i.e.

, f((p(s)) = g(<p(s)) for //-almost all * if /(*) = g(*) for /{-almost all s. Thus T may be

regarded as a

dition (ii) shows that T maps //-equivalent functions into //-equivalent functions, i.e.

, f((p(s)) = g(<p(s)) for //-almost all * if /(*) = g(*) for /{-almost all s. Thus T may be

regarded as a

**linear map**in the .F-spaee M(S) provided that Tf is //-measurable ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact