Linear Operators: General theory |
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Page 166
... assume that | f ( $ ) | ≤ 2 | f ( s ) | for all s in S. By the dominated conver- gence theorem , ƒn → ƒ in L „ ( μ ) , and hence { f } is a Cauchy sequence in Lu ) and fn ( ) } is a Cauchy sequence in L1 ( u ) . Since f ( ) is a μ ...
... assume that | f ( $ ) | ≤ 2 | f ( s ) | for all s in S. By the dominated conver- gence theorem , ƒn → ƒ in L „ ( μ ) , and hence { f } is a Cauchy sequence in Lu ) and fn ( ) } is a Cauchy sequence in L1 ( u ) . Since f ( ) is a μ ...
Page 177
Nelson Dunford, Jacob T. Schwartz. may be assumed that 2 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 2 is positive . Let ...
Nelson Dunford, Jacob T. Schwartz. may be assumed that 2 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 2 is positive . Let ...
Page 178
... assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is 2 - measurable there is a sequence { g } of simple functions converging to g ( s ) for every s in F except on a set ECF with v ( λ , E ) = 0 ( by Corollary 6.13 ( a ) ...
... assume that v ( u , F ) < ∞ and v ( 2 , F ) < ∞ . Since g is 2 - measurable there is a sequence { g } of simple functions converging to g ( s ) for every s in F except on a set ECF with v ( λ , E ) = 0 ( by Corollary 6.13 ( a ) ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ