## Linear Operators, Part 1 |

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Page 57

Q.E.D. - 2 THEOREM. A continuous

all of another has a continuous

a continuous

...

Q.E.D. - 2 THEOREM. A continuous

**linear**one-to-one**map**of one F-space ontoall of another has a continuous

**linear**inverse. PRoof. Let 3:, ) be F-spaces and Ta continuous

**linear**one-toone**map**with T3 = }). Since (T-1)-1 = T**maps**open sets...

Page 58

The

Hence, by Theorem 2, its inverse pra' is continuous. Thus T = propra' is

continuous (I.4.17). Q.E.D. 5 THEOREM. If a

each of two ...

The

**map**pry : [a, Taj--a of () onto 3 is one-to-one,**linear**, and continuous (I.8.3).Hence, by Theorem 2, its inverse pra' is continuous. Thus T = propra' is

continuous (I.4.17). Q.E.D. 5 THEOREM. If a

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

Page 490

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

Los O, 1], p > 1, to Los O, 1] has the form d 1. g(s) = #| K(s,t)f(t)dt, no satisfactory

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

are ...

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

**linear map**fromLos O, 1], p > 1, to Los O, 1] has the form d 1. g(s) = #| K(s,t)f(t)dt, no satisfactory

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

are ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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