## Linear Operators: General theory |

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Page 120

If we put t = al!qb~1,p we obtain the

; hence, the

)/!/!.. * = g(*)/|g|*. we find that It follows from Lemma 2.12, Theorem 2.19(a), ...

If we put t = al!qb~1,p we obtain the

**inequality**ab 5S a1>jp+bqlq, valid for a, b > 0; hence, the

**inequality**\ab\ ^ lal'/p + l^l*/? is valid for all scalars a, b. Putting a = /(*)/!/!.. * = g(*)/|g|*. we find that It follows from Lemma 2.12, Theorem 2.19(a), ...

Page 121

We observe that the

functions) the

This observation is obvious in the case of Minkowski's

...

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 L„.This observation is obvious in the case of Minkowski's

**inequality**. To see that it is...

Page 248

The above

follows from the postulates for that the Schwarz

zero. Hence suppose that X 0 t£ y. For an arbitrary complex number a 0 g (x+cty,

...

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 y iszero. Hence suppose that X 0 t£ y. For an arbitrary complex number a 0 g (x+cty,

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero