Linear Operators: General theory |
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Page 106
... simple functions . If for every E in with ( u , E ) < ∞ , the product Zef of f with the characteristic function E ... functions are M ( S , 2 , μ , X ) , M ( S , 2 , μ ) , M ( S ) and symbols besides TM ( S ) for the set of totally ...
... simple functions . If for every E in with ( u , E ) < ∞ , the product Zef of f with the characteristic function E ... functions are M ( S , 2 , μ , X ) , M ( S , 2 , 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 EeΣ the integral over E of a μ - integrable simple func- tion h is defined by the equation n Σαμ ( ΕΕ ; ) . [ _h ( s ) u ( ds ) = √ f ( s ) ...
... simple function " and " u - integrable simple function " will be used interchan- geably . For an EeΣ the integral over E of a μ - integrable simple func- tion h is defined by the equation n Σαμ ( ΕΕ ; ) . [ _h ( s ) u ( ds ) = √ f ( s ) ...
Page 322
... function f defined on S is u - simple if it is a finite linear combination of characteristic functions of sets in * ; this is evidently the case if and only if ƒ is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that f ...
... function f defined on S is u - simple if it is a finite linear combination of characteristic functions of sets in * ; this is evidently the case if and only if ƒ is 2 - simple . It follows from Corollaries III.6.13 and III.6.14 that f ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite 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 theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ