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Page 120
If we put t = alla b - 1 / P we obtain the inequality ab < app + bo / q , valid for a , b > 0 ; hence , the inequality | ab | < | a / p + 181 % a is valid for all scalars a , b . Putting a = f ( s ) / \ Ho , b = g ( s ) / | g | q ...
If we put t = alla b - 1 / P we obtain the inequality ab < app + bo / q , valid for a , b > 0 ; hence , the inequality | ab | < | a / p + 181 % a is valid for all scalars a , b . Putting a = f ( s ) / \ Ho , b = g ( s ) / | g | q ...
Page 121
To see that it is justified in case of Hölder's inequality , note that Hölder's inequality implies that 1g1 < lugl , if fel , ge L 1 1 + р 9 1 , and at least one of f and g is scalar III.3.4 121 LEBESGUE SPACES.
To see that it is justified in case of Hölder's inequality , note that Hölder's inequality implies that 1g1 < lugl , if fel , ge L 1 1 + р 9 1 , and at least one of f and g is scalar III.3.4 121 LEBESGUE SPACES.
Page 248
The above inequality , known as the Schwarz inequality , will be proved first . It follows from the postulates for H that the Schwarz inequality is valid if either x or y is zero . Hence suppose that 8C # 0 #y . For an arbitrary complex ...
The above inequality , known as the Schwarz inequality , will be proved first . It follows from the postulates for H that the Schwarz inequality is valid if either x or y is zero . Hence suppose that 8C # 0 #y . For an arbitrary complex ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc Proof properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero