Linear Operators: General theory |
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Page 98
... v ( E ) = lim { E } μ ( E ) - 6 LEMMA . The total variation of an additive set function defined on a field of ... Fi Σμ ( A ; ) ≤Σμ ( E ; ) | + Σ│μ ( F . ) ≤ v ( μ , E ) + v ( μ , F ) , and hence ( i ) v ( μ , E ○ F ) ≤ v ( μ , E ) ...
... v ( E ) = lim { E } μ ( E ) - 6 LEMMA . The total variation of an additive set function defined on a field of ... Fi Σμ ( A ; ) ≤Σμ ( E ; ) | + Σ│μ ( F . ) ≤ v ( μ , E ) + v ( μ , F ) , and hence ( i ) v ( μ , E ○ F ) ≤ v ( μ , E ) ...
Page 120
... fi and f2 lies in L ( S , E , μ ) , and f1 + f2p ≤ = ( c ) \ f1 - f2 \ p = 0 if and only if f - f2 is a null ... v ( μ , ds ) ≤ fs { f1 ( 8 ) | + \ f2 ( 8 ) } 3 v ( μ , ds ) = √§ \ f1 ( 8 ) | { \ 120 III.3.3 III . INTEGRATION AND SET ...
... fi and f2 lies in L ( S , E , μ ) , and f1 + f2p ≤ = ( c ) \ f1 - f2 \ p = 0 if and only if f - f2 is a null ... v ( μ , ds ) ≤ fs { f1 ( 8 ) | + \ f2 ( 8 ) } 3 v ( μ , ds ) = √§ \ f1 ( 8 ) | { \ 120 III.3.3 III . INTEGRATION AND SET ...
Page 171
... fi , f2 be positive functions in L ,. Show that f + f2 " f1 " + f2 " . 32 Find an example of a countably additive , bounded , vector valued function μ defined on a σ - field Σ of subsets of S for which v ( u , S ) + ∞o . 33 Show that ...
... fi , f2 be positive functions in L ,. Show that f + f2 " f1 " + f2 " . 32 Find an example of a countably additive , bounded , vector valued function μ defined on a σ - field Σ of subsets of S for which v ( u , S ) + ∞o . 33 Show that ...
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