Linear Operators: General theory |
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Page 40
... a field . Conversely , if R / I is a field , it contains no ideals and hence R has no ideals properly containing I. = If R is a ring with unit e , then an element a in R is called ( right , left ) regular in R in case R contains a ...
... a field . Conversely , if R / I is a field , it contains no ideals and hence R has no ideals properly containing I. = If R is a ring with unit e , then an element a in R is called ( right , left ) regular in R in case R contains a ...
Page 126
... a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a ...
... a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued additive set function defined on a ...
Page 136
... field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family 7. The intersection of all fields ...
... field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all subsets of S , which contains a given family 7. The intersection of all fields ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ