Linear Operators: General theory |
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Page 95
... Functions In contrast to the terms real function , complex function , etc. , where the adjectives real and complex refer to the range of the func- tion , the term set function is commonly used in mathematics for a function whose domain ...
... Functions In contrast to the terms real function , complex function , etc. , where the adjectives real and complex refer to the range of the func- tion , the term set function is commonly used in mathematics for a function whose domain ...
Page 103
... function or , when u is understood , simply a null function if the set S ( fa ) is a u - null set for each > 0 . It is important to observe that a null function with respect to a finitely additive set function need not vanish almost ...
... function or , when u is understood , simply a null function if the set S ( fa ) is a u - null set for each > 0 . It is important to observe that a null function with respect to a finitely additive set function need not vanish almost ...
Page 165
... function , it is evident that ƒ is also a μ - integrable simple function , and that f ( s ) μ ( ds ) = SE f ( s ) μ ( ds ) for E € 21. Let ƒ be a μ - integrable function and let { f } be a sequence of μ - integrable simple functions ...
... function , it is evident that ƒ is also a μ - integrable simple function , and that f ( s ) μ ( ds ) = SE f ( s ) μ ( ds ) for E € 21. Let ƒ be a μ - integrable function and let { f } be a sequence of μ - integrable simple 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