## Linear Operators: General theory |

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Page 120

If we put t = a1/s&-1/j) we obtain the

hence, the

/(*)/|/|p. * = g(*)/|g|«. we find that ^ - +- \g(*)\Q\g\\-°\f\,- V 1 It follows from Lemma ...

If we put t = a1/s&-1/j) we obtain the

**inequality**ab iS ap/p+65/g, valid for a, b > 0;hence, the

**inequality**\ab\ ^ |a |p/p + 16 |c/<? is valid for all scalars a, b. Putting a =/(*)/|/|p. * = g(*)/|g|«. we find that ^ - +- \g(*)\Q\g\\-°\f\,- V 1 It follows from Lemma ...

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 Lv.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 ^ (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 yis zero. Hence suppose that x ^ 0 y. For an arbitrary complex number a 0 ^ (x+cty,

...

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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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero