## Linear Operators: General theory |

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Page 120

\jsmg(*Mds)\ 5; |/|p|g|,. Proof. The function <p(t) = tp',p-\-t~''!iq has a positive

derivative for t > 1, and a negative derivative for 0 < t < 1. Hence, its minimum

value for I > 0 is 99(1) = 1. If we put t = al'"b~1,p we obtain the

-\-bqjq, ...

\jsmg(*Mds)\ 5; |/|p|g|,. Proof. The function <p(t) = tp',p-\-t~''!iq has a positive

derivative for t > 1, and a negative derivative for 0 < t < 1. Hence, its minimum

value for I > 0 is 99(1) = 1. If we put t = al'"b~1,p we obtain the

**inequality**ab ap!p-\-bqjq, ...

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 ^ 0 ^ y. For an arbitrary complex number a 0 g (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 ^ 0 ^ y. For an arbitrary complex number a 0 g (x+

xy, ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact