## Linear Operators: General theory |

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Results 1-3 of 73

Page 169

1

and a finite collection of disjoint sets A1 -d-neE such that Ax u . • • U ^n = v(/x, Ee)

< e, and sup \f(s)—f(t)\ < e, 7 = 1,..., n. ''t€A' 2 Let A C S be a fx-null set.

...

1

**Show**that / e TM(S, E, fx) if and only if for each e > 0 there exists a set Ec e Eand a finite collection of disjoint sets A1 -d-neE such that Ax u . • • U ^n = v(/x, Ee)

< e, and sup \f(s)—f(t)\ < e, 7 = 1,..., n. ''t€A' 2 Let A C S be a fx-null set.

**Show**that...

Page 358

Let En(x, y) = ^Ln<f>i(x)<f>i(y)-

projection S„ in each of the spaces LP, BV, CBV, AC, C(*', 1 ^ p ^ co, k = 0, 1, 2 co

.

...

Let En(x, y) = ^Ln<f>i(x)<f>i(y)-

**Show**that Snf is given by the formula and is aprojection S„ in each of the spaces LP, BV, CBV, AC, C(*', 1 ^ p ^ co, k = 0, 1, 2 co

.

**Show**that the range of Sn lies in C<"». 8**Show**that Sn -*-I strongly in any one of...

Page 360

21

system is localized if and only if max \En(x, y)\ ^ M, < oo for each e> 0. l*-v|s* 22

Suppose that (S„f)(x) -*□ f(x) uniformly for every / in AC.

...

21

**Show**that if | Jj En(x> x)dz\ ^ M, then the convergence of S„f for a given c.o.n.system is localized if and only if max \En(x, y)\ ^ M, < oo for each e> 0. l*-v|s* 22

Suppose that (S„f)(x) -*□ f(x) uniformly for every / in AC.

**Show**that there exists a...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 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

a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero