Linear Operators: General theory |
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Page 120
If we put t alla b - 1 / P we obtain the inequality ab app +69/9 , valid for a , b > 0 ; hence , the inequality | ab | = | a | p / p + 16 ° / q is valid for all scalars a , b . Putting a = f ( s ) / \ | lp , b = g ( s ) // g | q ...
If we put t alla b - 1 / P we obtain the inequality ab app +69/9 , valid for a , b > 0 ; hence , the inequality | ab | = | a | p / p + 16 ° / q is valid for all scalars a , b . Putting a = f ( s ) / \ | lp , b = g ( s ) // g | q ...
Page 121
Thus where no confusion will result , we shall speak simply of “ a function in Ly ” . In the sequel the symbol | rather than [ 1 ] will be used for an element in Lp . We observe that the inequality of Minkowski and in ...
Thus where no confusion will result , we shall speak simply of “ a function in Ly ” . In the sequel the symbol | rather than [ 1 ] will be used for an element in Lp . We observe that the inequality of Minkowski and in ...
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 x + 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 x + 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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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero