## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 75

Page 54

Since the

proof is complete . Q.E.D. We now ... A linear

another is continuous if and only if it

Proof .

Since the

**mapping**x + x / k , is a homeomorphism of X with itself , by 10 ( ii ) , theproof is complete . Q.E.D. We now ... A linear

**mapping**of one F - space intoanother is continuous if and only if it

**maps**bounded sets into bounded sets .Proof .

Page 55

1 , 2 , ... , and T is a x in X. Then lim oTqX O uniformly for n continuous linear

of X into Y. Proof . The linearity of T is an immediate consequence of the linearity

of the operators Tn . For each x , the sequence { T , x } is convergent , and ...

1 , 2 , ... , and T is a x in X. Then lim oTqX O uniformly for n continuous linear

**map**of X into Y. Proof . The linearity of T is an immediate consequence of the linearity

of the operators Tn . For each x , the sequence { T , x } is convergent , and ...

Page 513

12 If H is a Hilbert space , the

with either the uniform or weak operator topology . By considering the sequence (

An } defined in Exercise 11 , show that this

12 If H is a Hilbert space , the

**mapping**T → T * of B ( H ) into itself is continuouswith either the uniform or weak operator topology . By considering the sequence (

An } defined in Exercise 11 , show that this

**mapping**is not continuous in the ...### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 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

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero