Linear Operators: General theory |
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Page 126
... valued , complex valued , or extended real valued additive set function defined on a field Σ of subsets of a set S. Then u is said to be countably additive if 00 μ ( UE ) = u ( E ) i = 1 i = 1 Σ whenever E , Eg , ... are disjoint sets ...
... valued , complex valued , or extended real valued additive set function defined on a field Σ of subsets of a set S. Then u is said to be countably additive if 00 μ ( UE ) = u ( E ) i = 1 i = 1 Σ whenever E , Eg , ... are disjoint sets ...
Page 224
... valued . It will be assumed that the reader is familiar with the elementary theory of complex valued analytic functions of one complex variable and this theory will be applied here to obtain the extensions which will be used later ...
... valued . It will be assumed that the reader is familiar with the elementary theory of complex valued analytic functions of one complex variable and this theory will be applied here to obtain the extensions which will be used later ...
Page 238
... complex valued functions f , g defined on a specified domain S. Here , addition and multiplication are understood to ... valued functions with the stated properties , or of all complex valued functions with these properties . Unless the ...
... complex valued functions f , g defined on a specified domain S. Here , addition and multiplication are understood to ... valued functions with the stated properties , or of all complex valued functions with these properties . Unless the ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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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