## Linear Operators: General theory |

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Page 57

Q.E.D. 2 Theorem. A continuous

another has a continuous

continuous

onto ...

Q.E.D. 2 Theorem. A continuous

**linear**one-to-one**map**of one F-space onto all ofanother has a continuous

**linear**inverse. Proof. Let X, $ be F-spaces and T acontinuous

**linear**one-to- one**map**with TX = Since (T-1)-1 = T**maps**open setsonto ...

Page 490

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

Lp[0, 1], p > 1, to L„[0, 1] has the form d r1 g(«) = -j K(s,t)f(t)dt, as Jo no satisfactory

expression for the norm of T is known. No conditions on K(s, t) are known which ...

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

**linear map**fromLp[0, 1], p > 1, to L„[0, 1] has the form d r1 g(«) = -j K(s,t)f(t)dt, as Jo no satisfactory

expression for the norm of T is known. No conditions on K(s, t) are known which ...

Page 664

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

, f(q>{s)) = g(<p{s)) for ^-almost all s if f(s) = g(s) for (U-almost all s. Thus T may be

regarded as a

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

, f(q>{s)) = g(<p{s)) for ^-almost all s if f(s) = g(s) for (U-almost all s. Thus T may be

regarded as a

**linear map**in the F-space 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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