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 Σ with v ( u , E ) < ∞ , the product % f of ƒ with the characteristic function XE 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 Σ with v ( u , E ) < ∞ , the product % f of ƒ with the characteristic function XE of E is totally measurable , the ...
Page 119
... measurable . = 0 Next suppose that we consider a function f ( vector or extended real - valued ) which is defined ... functions we make no change in F ( S , Σ , μ , X ) , or in any of the theorems or lemmas of this section . Finally , suppose ...
... measurable . = 0 Next suppose that we consider a function f ( vector or extended real - valued ) which is defined ... functions we make no change in F ( S , Σ , μ , X ) , or in any of the theorems or lemmas of this section . Finally , suppose ...
Page 179
... measurable function defined on S and λ ( E ) = √ ̧ f ( s ) μ ( ds ) , ΕΕΣ . Let g be a non - negative 2 ... functions h for which the equation √2h ( s ) 2 ( ds ) = √ 2f ( s ) h ( s ) μ ( ds ) , E ΕΕΣ , is valid , then H clearly contains ...
... measurable function defined on S and λ ( E ) = √ ̧ f ( s ) μ ( ds ) , ΕΕΣ . Let g be a non - negative 2 ... functions h for which the equation √2h ( s ) 2 ( ds ) = √ 2f ( s ) h ( s ) μ ( ds ) , E ΕΕΣ , is valid , then H clearly contains ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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