Linear Operators: General theory |
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Page 106
... simple functions . If for every E in E with v ( u , E ) < ∞ , the product XEf off with the characteristic function ... functions are M ( S , Σ , μ , X ) , M ( S , Σ , μ ) , M ( S ) and symbols besides TM ( S ) for the set of totally ...
... simple functions . If for every E in E with v ( u , E ) < ∞ , the product XEf off with the characteristic function ... functions are M ( S , Σ , μ , X ) , M ( S , Σ , μ ) , M ( S ) and symbols besides TM ( S ) for the set of totally ...
Page 108
... simple function " and " u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation n | _ h ( s ) u ( ds ) = [ _ f ( s ) μu ( ds ) ...
... simple function " and " u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation n | _ h ( s ) u ( ds ) = [ _ f ( s ) μu ( ds ) ...
Page 322
... function ƒ defined on S is μ - simple if it is a finite linear combination of characteristic functions of sets in 2 * ; this is evidently the case if and only if f is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that ƒ ...
... function ƒ defined on S is μ - simple if it is a finite linear combination of characteristic functions of sets in 2 * ; this is evidently the case if and only if f is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that ƒ ...
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