## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 81

Page 334

By Corollary III.6.13(b), V, (a) → V(a) and so it follows that V is continuous at a = 0

. Since |W(w, s) ~ 2V(r, s), ... The fact that these functions are in M(S, 2, u) or in TM

(S, X, u)

By Corollary III.6.13(b), V, (a) → V(a) and so it follows that V is continuous at a = 0

. Since |W(w, s) ~ 2V(r, s), ... The fact that these functions are in M(S, 2, u) or in TM

(S, X, u)

**follows from Lemmas**III.2.11 and III.2.12. It is also clear that the value ...Page 421

Then the linear functionals on 3 which are continuous in the T topology are

precisely the functionals in T. The proof of Theorem 9 will be based on the

linear space 3., ...

Then the linear functionals on 3 which are continuous in the T topology are

precisely the functionals in T. The proof of Theorem 9 will be based on the

**following lemma**. 10 LEMMA. If g, fi, ..., f, are any n +1 linear functionals on alinear space 3., ...

Page 422

By Theorem 2.10, there exists a nonzero linear T-continuous functional g, and a

real constant c, such that &g($),) s c. By Lemma 1.11, g(\\,) = 0; i.e., f(a) = 0

implies g(a)=0. It

By Theorem 2.10, there exists a nonzero linear T-continuous functional g, and a

real constant c, such that &g($),) s c. By Lemma 1.11, g(\\,) = 0; i.e., f(a) = 0

implies g(a)=0. It

**follows from Lemma**10 that g = xf for some non-zero scalar 2.### What people are saying - Write a review

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

### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

### Other editions - View all

### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero