## Linear Operators: General theory |

### From inside the book

Results 1-3 of 84

Page 120

If we put t = allqb~1,p we obtain the

hence, the

\f\„ b = g(s)l\g\Q, we find that \f(s)g(S)\ g - \m\p\f\\-'\g\q +" \i(»)\q\gV\f\,- V 9 It follows

...

If we put t = allqb~1,p we obtain the

**inequality**ab ^ ap/p-\-bq/q, valid for a, b > 0;hence, the

**inequality**|a6| ;£ H'/P + M8/? is valid for all scalars a, b. Putting a = f(s)l\f\„ b = g(s)l\g\Q, we find that \f(s)g(S)\ g - \m\p\f\\-'\g\q +" \i(»)\q\gV\f\,- V 9 It follows

...

Page 121

In the sequel the symbol / rather than [/] will be used for an element in Lp. We

observe that the

functions) the

This ...

In the sequel the symbol / rather than [/] will be used for an element in Lp. We

observe that the

**inequality**of Minkowski and (in the case of scalar valuedfunctions) the

**inequality**of Holder may be regarded as applying to the spaces Lp.This ...

Page 248

The above

follows from the postulates for § that the Schwarz

is zero. Hence suppose that x 9^ 0 ^ y. For an arbitrary complex number a 0 ^ (x+

xy, ...

The above

**inequality**, known as the Schwarz**inequality**, will be proved first. Itfollows from the postulates for § that the Schwarz

**inequality**is valid if either x or yis zero. Hence suppose that x 9^ 0 ^ y. For an arbitrary complex number a 0 ^ (x+

xy, ...

### 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 | |

78 other sections not shown

### Other editions - View all

### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations disjoint Doklady Akad Duke Math element ergodic theorem exists finite dimensional function defined Hausdorff space Hence Hilbert space homeomorphism ibid inequality integral Lebesgue Lebesgue measure Lemma linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc Proof properties proved real numbers reflexive Riesz Russian Sbornik N. S. scalar self-adjoint semi-group sequentially compact Show simple functions spectral Studia Math subset subspace Suppose theory topological space Trans uniformly Univ valued function Vber vector space weak topology weakly compact zero