## Linear Operators: General theory |

### From inside the book

Results 1-3 of 83

Page 289

there is a functional y* e L* such that x**{x*) = y*(g) when g and x* are connected,

as in Theorem 1, by the formula x*f = jsf(s)g(s)p(ds), feLv. Applying Theorem 1 ...

**Proof**. Let x** e (!<*)*. By Theorem 1, L* is isometrically isomorphic to LQ, so thatthere is a functional y* e L* such that x**{x*) = y*(g) when g and x* are connected,

as in Theorem 1, by the formula x*f = jsf(s)g(s)p(ds), feLv. Applying Theorem 1 ...

Page 415

Q.E.D. Since the commutativity of the group G is not essential to the

same result holds for non-Abelian topological groups. 4 Lemma. For arbitrary

sets A, B in a linear space (i) co(xA) =aco(A), co(A+B) = co(A)+co(B). // X is a

linear ...

Q.E.D. Since the commutativity of the group G is not essential to the

**proof**, thesame result holds for non-Abelian topological groups. 4 Lemma. For arbitrary

sets A, B in a linear space (i) co(xA) =aco(A), co(A+B) = co(A)+co(B). // X is a

linear ...

Page 434

and proceed as in the first part of the

a subsequence {«/„,} of {xn} such that ^imm^^, x*ym exists for each x* in the set H

of that

and proceed as in the first part of the

**proof**of the preceding theorem to constructa subsequence {«/„,} of {xn} such that ^imm^^, x*ym exists for each x* in the set H

of that

**proof**. Let Km = co{ym, ym+1,. . .} and let y0 be an arbitrary point in D „=1 ...### 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 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