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 a > 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 a > 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 is u - measurable . If f is a μ - integrable simple function , it is evident that f is also a μ - integrable simple function , and that SÅ f ( s ) μ ( ds ) = SE f ( s ) μ ( ds ) for E e 21. Let f be a μ - integrable function ...
... function is u - measurable . If f is a μ - integrable simple function , it is evident that f is also a μ - integrable simple function , and that SÅ f ( s ) μ ( ds ) = SE f ( s ) μ ( ds ) for E e 21. Let f be a μ - integrable function ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ