## Linear Operators: General theory |

### From inside the book

Results 1-3 of 77

Page 279

Hence ( i ) ( Hf ) ( t ) = f ( h ( t ) ) , fe C ( S ) , which

the H is continuous . To see that h is continuous , let N be a neighborhood of the

point 80 = h ( to ) . By Theorems 1.5.2 and 1.5.9 there is a continuous function f ...

Hence ( i ) ( Hf ) ( t ) = f ( h ( t ) ) , fe C ( S ) , which

**shows**that \ H || = ||| and provesthe H is continuous . To see that h is continuous , let N be a neighborhood of the

point 80 = h ( to ) . By Theorems 1.5.2 and 1.5.9 there is a continuous function f ...

Page 661

Since { A ( n ) } is bounded , Lemma II.3.30

{ A ( n ) x } is weakly sequentially compact is a closed linear manifold . The

identity Tn ( * ) n - 1 An ) - An - 1 ) n n

by ...

Since { A ( n ) } is bounded , Lemma II.3.30

**shows**that the set of those x for which{ A ( n ) x } is weakly sequentially compact is a closed linear manifold . The

identity Tn ( * ) n - 1 An ) - An - 1 ) n n

**shows**that { T " / n } is bounded and hence ,by ...

Page 665

Since convergence in L , implies convergence in measure ( III.3.6 ) , Lemma 6

Tf = g and T is closed . By the closed graph theorem ( II.2.4 ) T is continuous and

...

Since convergence in L , implies convergence in measure ( III.3.6 ) , Lemma 6

**shows**that Tin T / in measure if In → f in Lp . Thus if In L , and Tfn → g in L , thenTf = g and T is closed . By the closed graph theorem ( II.2.4 ) T is continuous and

...

### 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 |

quences | 26 |

Copyright | |

80 other sections not shown

### Other editions - View all

### Common terms and phrases

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