Linear Operators: General theory |
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Page 175
... ΕΕΣ . Then , is a μ - continuous finite non - negative measure on E. To com- plete the proof we demonstrate that 2 ... ΕΕΣ , and then , a fortiori , ( ii ) μ ( EA ) — k21 ( E ) ≤ 0 , μ ( EA ' ) — kλ ( EA ′ ) ≥ 0 , ΕΕΣ . Therefore 1 ...
... ΕΕΣ . Then , is a μ - continuous finite non - negative measure on E. To com- plete the proof we demonstrate that 2 ... ΕΕΣ , and then , a fortiori , ( ii ) μ ( EA ) — k21 ( E ) ≤ 0 , μ ( EA ' ) — kλ ( EA ′ ) ≥ 0 , ΕΕΣ . Therefore 1 ...
Page 179
... ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and √2g ( s ) 2 ( ds ) = √2f ( $ ) g ( s ) μ ( ds ) , E ΕΕΣ . PROOF . The μ - measurability of fg follows from Lemma 3. If we let H be ...
... ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and √2g ( s ) 2 ( ds ) = √2f ( $ ) g ( s ) μ ( ds ) , E ΕΕΣ . PROOF . The μ - measurability of fg follows from Lemma 3. If we let H be ...
Page 181
... ΕΕΣ . PROOF . Using the Radon - Nikodým theorem ( Theorem 2 ) we find ( u ) -integrable functions g and h such that Since it follows that E λ ( E ) = √g ( s ) v ( μ , ds ) , ΕΕΣ , ΕΕΣ . μ ( E ) = √2 h ( s ) v ( μ , ds ) , E v ( μ , E ) ...
... ΕΕΣ . PROOF . Using the Radon - Nikodým theorem ( Theorem 2 ) we find ( u ) -integrable functions g and h such that Since it follows that E λ ( E ) = √g ( s ) v ( μ , ds ) , ΕΕΣ , ΕΕΣ . μ ( E ) = √2 h ( s ) v ( μ , ds ) , E v ( μ , E ) ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ