Linear Operators: General theory |
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Page 166
... assume that f ( 8 ) | ≤ 2 | f ( 8 ) | for all s in S. By the dominated conver- gence theorem , fn → f in L , ( u ) , and hence { f } is a Cauchy sequence in L „ ( μ ) and { \ ƒn ( · ) P } is a Cauchy sequence in L1 ( u ) . Since fn ...
... assume that f ( 8 ) | ≤ 2 | f ( 8 ) | for all s in S. By the dominated conver- gence theorem , fn → f in L , ( u ) , and hence { f } is a Cauchy sequence in L „ ( μ ) and { \ ƒn ( · ) P } is a Cauchy sequence in L1 ( u ) . Since fn ...
Page 177
Nelson Dunford, Jacob T. Schwartz. may be assumed that λ is real valued . A real valued set function can be represented as the difference of its positive and negative variations ( 4.11 ) and so we may also assume that λ is positive . Let ...
Nelson Dunford, Jacob T. Schwartz. may be assumed that λ is real valued . A real valued set function can be represented as the difference of its positive and negative variations ( 4.11 ) and so we may also assume that λ is positive . Let ...
Page 675
... assume that f≥ 0. Let a ' be the com- plement of the set a = { s | f ( s ) ≥ 1 } . Since fe L , it follows that μ ... assumed that ƒ is in L1 . In view of Lemma 4 it may also be assumed that T is positive . Let e∞ = 100 { s sup A ( T ...
... assume that f≥ 0. Let a ' be the com- plement of the set a = { s | f ( s ) ≥ 1 } . Since fe L , it follows that μ ... assumed that ƒ is in L1 . In view of Lemma 4 it may also be assumed that T is positive . Let e∞ = 100 { s sup A ( T ...
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