Linear Operators: General theory |
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Page 108
... u - integrable if it differs by a null function from a function of the form n f = Σ xil Ei i i = 1 Σ where Ef1 ( x¿ ) , i = 1 , . . . , n , are disjoint sets in with union S and where x = 0 if v ( u , E ; ) = ∞o . The phrases " u - ...
... u - integrable if it differs by a null function from a function of the form n f = Σ xil Ei i i = 1 Σ where Ef1 ( x¿ ) , i = 1 , . . . , n , are disjoint sets in with union S and where x = 0 if v ( u , E ; ) = ∞o . The phrases " u - ...
Page 112
... integrable on S if there is a sequence { f } of u - integrable simple functions converging to fin u - measure on S and satisfying in addition the equation lim f ̧ \ fm ( 3 ) —fn ... u 112 III.2.17 III . INTEGRATION AND SET FUNCTIONS.
... integrable on S if there is a sequence { f } of u - integrable simple functions converging to fin u - measure on S and satisfying in addition the equation lim f ̧ \ fm ( 3 ) —fn ... u 112 III.2.17 III . INTEGRATION AND SET FUNCTIONS.
Page 180
... integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( ) g ( ) is μ - integrable . Theorem 4 applies in the same manner to prove the 2 - ...
... integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( ) g ( ) is μ - integrable . Theorem 4 applies in the same manner to prove the 2 - ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ