Linear Operators: General theory |
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Page 106
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in with v ( u , E ) < ∞ , the product Zef of f with the characteristic function ZE of E is totally measurable , the ...
... measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E in with v ( u , E ) < ∞ , the product Zef of f with the characteristic function ZE of E is totally measurable , the ...
Page 119
... measurable . Next suppose that we consider a function ƒ ( vector or extended real - valued ) which is defined only ... functions we make no change in F ( S , Z , μ , X ) , or in any of the theorems or lemmas of this section . Finally , ...
... measurable . Next suppose that we consider a function ƒ ( vector or extended real - valued ) which is defined only ... functions we make no change in F ( S , Z , μ , X ) , or in any of the theorems or lemmas of this section . Finally , ...
Page 663
... functions but classes of equivalent functions , it is seen that T may not be regarded as being defined on L , ( S ... measurable functions into u - measurable functions and u - equivalent functions into p - equivalent functions ...
... functions but classes of equivalent functions , it is seen that T may not be regarded as being defined on L , ( S ... measurable functions into u - measurable functions and u - equivalent functions into p - equivalent functions ...
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