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 Zef of ƒ with the characteristic function ΧΕ 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 Zef of ƒ with the characteristic function ΧΕ of E is totally measurable , the ...
Page 119
... measurable . Next suppose that we consider a function f ( vector or extended real - valued ) which is defined only ... functions we make no change in F ( S , E , μ , X ) , or in any of the theorems or lemmas of this section . Finally , ...
... measurable . Next suppose that we consider a function f ( vector or extended real - valued ) which is defined only ... functions we make no change in F ( S , E , 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 μ - measurable functions and u - equivalent functions into μ - 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 μ - measurable functions and u - equivalent functions into μ - equivalent functions ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ