Linear Operators: General theory |
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Page 95
... valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the integration process is defined with respect to a scalar valued set function u , the resulting integral v may be a vector valued set function . It is ...
... valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the integration process is defined with respect to a scalar valued set function u , the resulting integral v may be a vector valued set function . It is ...
Page 318
... Vector Valued Measures The theorems on spaces of set functions proved in the last section permit us to develop a more satisfactory theory of vector valued count- ably additive set functions ( briefly , vector valued measures ) than we ...
... Vector Valued Measures The theorems on spaces of set functions proved in the last section permit us to develop a more satisfactory theory of vector valued count- ably additive set functions ( briefly , vector valued measures ) than we ...
Page 319
... valued measures is uniformly countably additive on 1 , which contradicts the assumption that yu ( E ) > ɛ , m = 1 , 2 , . . .. This proves the first assertion . If μ ... vector valued IV.10.2 319 VECTOR VALUED MEASURES Vector Valued Measures.
... valued measures is uniformly countably additive on 1 , which contradicts the assumption that yu ( E ) > ɛ , m = 1 , 2 , . . .. This proves the first assertion . If μ ... vector valued IV.10.2 319 VECTOR VALUED MEASURES Vector Valued Measures.
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