Linear Operators: General theory |
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Page 164
... restriction to is in ca ( S , E ) . We define 22 ( E ) 2 ( E ) —λ ( E ) , E e E ; clearly λ2 ≥0 . If λ is not purely finitely additive , there is a non - zero λ ' e ca ( S , E ) such that ' ≤λ - 1 ; hence 11≤ + and supuec ( S ) = 21 ...
... restriction to is in ca ( S , E ) . We define 22 ( E ) 2 ( E ) —λ ( E ) , E e E ; clearly λ2 ≥0 . If λ is not purely finitely additive , there is a non - zero λ ' e ca ( S , E ) such that ' ≤λ - 1 ; hence 11≤ + and supuec ( S ) = 21 ...
Page 166
... restriction of μ to a subfield of Σ there is an- other type of restriction of common occurrence in integration theory . In the following discussion of this other type of restriction it is not necessary to assume that μ is non - negative ...
... restriction of μ to a subfield of Σ there is an- other type of restriction of common occurrence in integration theory . In the following discussion of this other type of restriction it is not necessary to assume that μ is non - negative ...
Page 168
... restriction μ1 of μ to Σ has the following properties : ( i ) the measure space ( S1 , E1 , M1 ) is o - finite ; ( ii ) the B - space L ( S1 , E1 , 1 , 1 ) is separable ; ( iii ) GCL ( S1 , E1 , 1 , 1 ) . = 1 , 2 , be PROOF . Let { f } ...
... restriction μ1 of μ to Σ has the following properties : ( i ) the measure space ( S1 , E1 , M1 ) is o - finite ; ( ii ) the B - space L ( S1 , E1 , 1 , 1 ) is separable ; ( iii ) GCL ( S1 , E1 , 1 , 1 ) . = 1 , 2 , be PROOF . Let { f } ...
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